For more information, please see full course syllabus of AP Calculus AB
For more information, please see full course syllabus of AP Calculus AB
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The Chain Rule
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- Intro
- The Chain Rule
- Example I: Identify f(x) and g(x) and Differentiate
- Example II: Identify f(x) and g(x) and Differentiate
- Example III: Differentiate
- Example IV: Differentiate f(x) = -5 / (x² + 3)³
- Example V: Differentiate f(x) = cos(x² + c²)
- Example VI: Differentiate f(x) = cos⁴x +c²
- Example VII: Differentiate
- Example VIII: Differentiate f(x) = sin(tan x²)
- Example IX: Differentiate f(x) = sin(tan² x)
- Intro 0:00
- The Chain Rule 0:13
- Recall the Composite Functions
- Derivatives of Composite Functions
- Example I: Identify f(x) and g(x) and Differentiate 6:41
- Example II: Identify f(x) and g(x) and Differentiate 9:47
- Example III: Differentiate 11:03
- Example IV: Differentiate f(x) = -5 / (x² + 3)³ 12:15
- Example V: Differentiate f(x) = cos(x² + c²) 14:35
- Example VI: Differentiate f(x) = cos⁴x +c² 15:41
- Example VII: Differentiate 17:03
- Example VIII: Differentiate f(x) = sin(tan x²) 19:01
- Example IX: Differentiate f(x) = sin(tan² x) 21:02
AP Calculus AB Online Prep Course
Transcription: The Chain Rule
Hello, welcome back to www.educator.com and welcome back to AP Calculus.0000
Today, we are going to discuss the chain rule.0005
Profoundly important, let us dive right on in.0008
Let us start off by recalling this idea of a composite function.0014
Recall composite functions.0018
Something like, if we had f(x) is equal to x³ and if we had g(x) is equal to, let us say, x² + 1,0030
the composite function f(g) is defined as f(g) of x.0040
We are basically, wherever we see x in here, we are just putting in the entire function.0047
It ends up being x² + 1³, that is it.0054
What we want to do, the chain rule, it allows you to find the derivative of a composite function.0060
That is it, that is all it is.0068
The majority of the functions that you are dealing with are actually going to be composite functions.0070
They are not just going to be simple functions like sin(x) or 5x or x².0073
They are going to be composite functions like x² + 1³ or sin(x)² + tan² x¹/5.0078
Most of them are going to be composites.0087
The chain rule is the one thing that you are going to be using over and over again.0088
We are going to be spending a fair amount of time with this.0093
That is all it is.0096
Let me write this down.0101
The chain rule allows us to find the derivatives of composite functions.0106
I apologize for my slightly sloppy writing.0117
Derivatives of composite functions, that is it, that is all it is.0120
Let us go ahead and go to blue.0132
If f and g, I will leave off the x, are differentiable, and F = f(g) or F = the composite of f(g), then, the composite is differentiable.0171
F’ is equal to f’ evaluated at g(x) × g’(x), that is the formula.0183
This is the formula for the chain rule.0195
It says that if I have a composite function, I take the derivative of the outer function.0197
I evaluate at g(x), or I put g(x) into the derivative of the function.0202
And then, I multiply it by the derivative of what is inside the derivative of g’.0210
This is why they call it the chain rule.0215
It makes more sense when you actually see the example.0217
You just keep differentiating from the outside in until you end up running out of things to differentiate.0220
That is why they call it the chain rule.0227
You do the outside function then the inside function, then the next function.0229
If you had f, g, h, you would do f’, you would do g’, then you do h’.0233
Again, this is the definition, all of this will make sense when we do the examples.0239
In the notation of this dy dx, the notation that we sometimes use, this is expressed this way.0246
Let y = some function of the variable u and u = some function of the variable x.0270
Then, dy dx is equal to, y is equal to function of u.0287
U was equal to a function of x, that means y is actually equal to a function of x.0297
When I put this u into here, I get g(x).0301
I want to find dy dx.0306
Dy dx is equal to dy du × du dx.0307
The nice thing about this particular notation is that you get the sort of see that this du and this du,0316
because we are treating them like quotients, you get to see that they cancel, leaving you with dy dx.0324
Again, it is just another way of looking at it.0332
I personally prefer this but that just a personal choice.0335
Note carefully.0341
When we say that f’(x) is equal to f’ evaluated at g(x) × g’(x), this is f’ at g(x).0349
When we have the specific x that we are talking about, we actually have to put in g(x).0368
We form f’, we find g(x), we put that in here.0373
We evaluate f’ at whatever number we get here.0378
And then, we multiply by g’ of whatever that number is.0381
This is very important.0385
Let us just go ahead and start with the examples.0391
I think once you see one or two examples, everything is going to make sense.0394
Let us do that.0397
Let f(x) = x³ + 5⁵, identify f(x) and g(x), then go ahead and differentiate.0403
In this particular case, f is going to be x⁵, the outer function.0413
G(x), the inner function, x³ + 5.0425
I should have put F, sorry.0433
That is okay, you will understand what is going on here.0438
F(x) = x⁵, this should probably be, let us go ahead and make this capital because f(x) and f(x), there might be some confusion.0440
I know that there is no confusion, but just for the sake of keeping it straight.0452
F(x) = x⁵, g(x) = x³ + 5.0457
You form your composite function.0461
F = f(g) which is equal to f(g), that is that.0463
Now we go ahead and differentiate.0473
We have f’(x), we differentiate from the outside, in.0476
We said that it is equal to, let us do the formula.0483
F’ at g(x) × g’(x).0493
For this particular function, f’, this is x⁵.0500
This thing⁵, we bring it down.0506
We treat it like a regular power rule.0510
5x³ + 5⁴, this is f’ at g(x).0512
F’ is 5 × whatever it is, the 4th, the g(x) is this one right here, × g’(x) what is inside.0521
Now, I take the derivative of what is inside, 3x².0531
That is my answer, f’(x).0536
I simplify, 3 × 5 is 15, x² × x³ + 5⁴.0540
That is what I’m doing, I’m just differentiating from the outside, in.0550
I have x³ + 5⁵.0558
I take care of the power rule first, x³ + 5⁴.0563
And then, I take the derivative of what is inside, 3x².0568
I keep going if I need to.0573
At this point, that is it, this is the last function that is quote inside.0574
I’m fine, I’m left with 15x² × x³ + 5⁴.0579
The sin of tan(x), identify f(x) and g(x), and then differentiate.0590
In this particular case, f is going to equal the sin(x) and g is equal to the tan(x).0594
The outside function is sin, the inside function is tan.0604
F’(x), this f’, the derivative of this, I take the derivative of the sin function which is cos of tan(x).0610
That stays.0624
I multiply by the derivative of what is inside, the derivative of tan is sec² x.0625
That is my final answer, if I want, I can go ahead and bring that forward.0633
Sec² x × the cos of tan(x), either one of these is fine.0639
You are working from the outside, in.0648
You take the derivative of sin, leaving what is inside alone.0650
And then, you multiply it by the derivative of what is inside and you keep going inside until you run out of things that are inside.0654
Let us see what we have got here.0665
Differentiate f(x) = ∛2x³ – 4x + 7.0672
I’m going to rewrite this with a rational exponent.0678
I’m going to write 2x³ – 4x + 7¹/3.0682
I differentiate my outside function is something to the 1/3.0693
I treat it as a regular power rule, then, I differentiate what is inside.0696
F’(x) = 1/3, I will leave what is inside alone.0700
2x³ – 4x + 7, until I finish with that.0708
This is that -1, × the derivative of what is inside, × 6x² – 4.0713
I’m done, I can stop there.0724
Just start from the outside and work your way in.0729
Differentiate f(x) = -5/ x² + 3³.0737
Now we have a quotient rule and the denominator happens to be something that is a composite function.0743
The derivative of that, we are going to use the quotient rule over the entire thing.0750
The chain rule for this one.0754
F’(x), this × the derivative of that - that × the derivative of this/ this².0758
This × the derivative of that.0765
We have x² + 3³ × the derivative of -5 is 0 - -5 × the derivative of this.0767
The derivative of this is, we do power rule first, 3 × x² + 3² × the derivative of what is inside, the derivative of x² + 3 is × 2x.0779
All of that /x² + 3³².0794
It is just 3 × 2.0800
This is going to be x² + 3⁶.0801
This is 0, I get 2 × 3 is 6, 6 × 5 is 30, - × - is +.0805
I get 30 × x² + 3²/ x² + 3⁶.0815
That goes with that, leaving 4.0825
I get 30/ x² + 3⁴.0828
That is it.0836
When I take the derivative of this, it is the outside 3, leave what is inside alone, finish that part.0838
And then, multiply by the derivative of what is inside.0846
That is the chain rule, it is a chain of functions.0849
If you have f and g, you are going to have two, one term and two terms.0853
If you have f, g, and h, you are going to have one term, two terms, three terms.0859
However many functions are nested inside of each other, that is how many terms you have in your final derivative expression.0864
Let us go to the next page, differentiate f(x) = cos (x)² + c².0875
f'(x) is equal to0881
C is a constant, c² is a constant, this is not a variable.0887
Do not let it throw you, x is the variable, f(x), x, everything else is treated as a constant.0893
F’(x), let us see, we have cos.0905
The derivative of cos is –sin.0911
We have –sin, we leave what is inside alone, x² + c² × the derivative of what is inside which is just 2x.0913
The derivative of c² is 0, it is constant.0923
Our final answer is, I will go ahead and bring this forward.0926
-2x × sin(x)² + c².0930
There you go.0937
F(x) = cos⁴ x + c².0944
F’(x) here, this, I’m going to write, cos⁴ x.0951
If you are comfortable with the notion of the 4 staying there, that is fine.0961
If you want, you can go ahead and rewrite this as cos(x), the whole thing⁴ +,0965
sorry, this is f(x), I’m rewriting it, + c².0973
Now we can do f’(x).0979
The derivative of this is 4 × cos(x)³.0982
Now, × the derivative of what is inside, the derivative of the cos is –sin, + the derivative of this is just 0.0990
Our final answer is -4 sin x cos³ x.0999
I went ahead and put the 3 back there.1008
X³ × e ⁻x³.1026
This is going to be product rule and chain rule.1030
Our chain rule here because this is e ⁺u, u is –x³.1035
It is a function of x itself.1039
F’(x), the derivative of this × this.1042
3x² × e ⁻x³ + the derivative of this × this.1049
The derivative of this is going to be e ⁻x³ × the derivative of this.1059
The derivative of this thing is e ⁺u du dx.1072
In other words, due to differentiation, this is inside.1079
The outside function is this.1083
This is the inside function.1086
The derivative of this is -3x².1087
This is the derivative of that part, now we have to multiply it by x³.1094
The derivative of this × this + the derivative of this × this.1102
The derivative of this is this whole thing.1107
Our final answer is 3x² e ⁻x³ – 3x⁵ e ⁻x³.1109
If you want to factor out the e ⁻x³, that is fine, you can.1123
But you are absolutely welcome to leave it as is.1125
Let us see, where are we?1131
I think that is correct, good.1138
Let us go to example 8, differentiate f(x) = sin of tan(x²).1141
Now we have three functions.1147
We have the sin, we have the tan, and we have x², which means we are going to have three terms.1149
We are going to work from outside, in, in our chain.1156
Do not worry, we are going to do plain more examples after this lesson is complete.1163
The derivative of the outside function, the derivative of sin is cos.1170
This is cos of tan(x)² × the derivative of tan(x)² is equal to sec² (x)².1173
This is still a function of x.1191
I take the derivative of that × 2x.1195
There you go.1200
My final answer is going to be 2x, cos of tan(x)² × sec² (x)².1204
Either one of these, it is absolutely fine.1220
Let us see what we have got.1225
See how that happen, there are three terms.1230
There are three functions, the sin, the tan, and the x².1232
These are all functions of x.1235
I have one term, two terms, three terms.1238
Keep going until you ran out.1241
In this particular case, my function of x is just x.1246
The derivative of x is just 1.1249
It is true, when we do keep going, that final derivative is just 1 so we do not put it there.1253
Now we have sin of tan² (x).1263
F’(x) = the derivative of sin is cos.1267
This is the cos of tan² (x).1274
Let me rewrite this again until we are comfortable with these ideas.1284
Let me rewrite f(x) = sin of tan(x)², three functions.1291
I have the sin function, I have the tan function, and I have the squared function.1311
Sin function, tan function, squared function.1319
Three functions, three terms.1324
F’(x) = the derivative of sin is cos.1328
I leave what is inside alone, the tan(x)².1332
Now I multiply by the derivative of what is inside.1338
The tan(x)² is equal to 2 × tan(x)¹ × the derivative of tan x which is sec² x.1341
My final answer, if I were to put it together, I can leave it like this.1359
I can bring the 2 to front, I guess.1364
2 × cos of tan² x × tan x × sec² x.1366
I hope that make sense.1383
Just the last one, at least for this particular lesson.1391
Here we have a product rule and we are going to have chain rule here and chain rule here.1395
F’(x) is equal to the derivative of this × this.1402
The derivative of this is 4 × x² – 2³ × the derivative of what is inside × 2x × this,1408
which is 5x² – 2x – 2⁸, + the derivative of this which is 8 × 5x² – 2x – 2⁷.1423
We are still taking the derivative of this × the derivative of what is inside which is tan x – 2 × this x² – 2⁴.1442
We have two functions, the outer function and the inner function, there are two terms.1463
The derivative of this × this + the derivative of this which is two terms × this.1468
There you go, you can stop there.1476
You do not really need to simplify this.1478
It is going to be too complex to simplify.1479
That is all, that is the chain rule.1482
Do not worry about this, the chain rule is very important.1484
In the next lesson, we are going to be continuing our example problems for the chain rule.1488
Thank you so much for joining us here at www.educator.com.1493
We will see you next time, bye.1496
Raffi Hovasapian
The Chain Rule
Slide Duration:Table of Contents
42m 8s
- Intro0:00
- Overview & Slopes of Curves0:21
- Differential and Integral0:22
- Fundamental Theorem of Calculus6:36
- Differentiation or Taking the Derivative14:24
- What Does the Derivative Mean and How do We Find it?15:18
- Example: f'(x)19:24
- Example: f(x) = sin (x)29:16
- General Procedure for Finding the Derivative of f(x)37:33
50m 53s
- Intro0:00
- Slope of the Secant Line along a Curve0:12
- Slope of the Tangent Line to f(x) at a Particlar Point0:13
- Slope of the Secant Line along a Curve2:59
- Instantaneous Slope6:51
- Instantaneous Slope6:52
- Example: Distance, Time, Velocity13:32
- Instantaneous Slope and Average Slope25:42
- Slope & Rate of Change29:55
- Slope & Rate of Change29:56
- Example: Slope = 233:16
- Example: Slope = 4/334:32
- Example: Slope = 4 (m/s)39:12
- Example: Density = Mass / Volume40:33
- Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change47:46
59m 12s
- Intro0:00
- Example I: Water Tank0:13
- Part A: Which is the Independent Variable and Which is the Dependent?2:00
- Part B: Average Slope3:18
- Part C: Express These Slopes as Rates-of-Change9:28
- Part D: Instantaneous Slope14:54
- Example II: y = √(x-3)28:26
- Part A: Calculate the Slope of the Secant Line30:39
- Part B: Instantaneous Slope41:26
- Part C: Equation for the Tangent Line43:59
- Example III: Object in the Air49:37
- Part A: Average Velocity50:37
- Part B: Instantaneous Velocity55:30
18m 43s
- Intro0:00
- Desmos Tutorial1:42
- Desmos Tutorial1:43
- Things You Must Learn To Do on Your Particular Calculator2:39
- Things You Must Learn To Do on Your Particular Calculator2:40
- Example I: y=sin x4:54
- Example II: y=x³ and y = d/(dx) (x³)9:22
- Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}13:15
51m 53s
- Intro0:00
- The Limit of a Function0:14
- The Limit of a Function0:15
- Graph: Limit of a Function12:24
- Table of Values16:02
- lim x→a f(x) Does not Say What Happens When x = a20:05
- Example I: f(x) = x²24:34
- Example II: f(x) = 727:05
- Example III: f(x) = 4.530:33
- Example IV: f(x) = 1/x34:03
- Example V: f(x) = 1/x²36:43
- The Limit of a Function, Cont.38:16
- Infinity and Negative Infinity38:17
- Does Not Exist42:45
- Summary46:48
24m 43s
- Intro0:00
- Example I: Explain in Words What the Following Symbols Mean0:10
- Example II: Find the Following Limit5:21
- Example III: Use the Graph to Find the Following Limits7:35
- Example IV: Use the Graph to Find the Following Limits11:48
- Example V: Sketch the Graph of a Function that Satisfies the Following Properties15:25
- Example VI: Find the Following Limit18:44
- Example VII: Find the Following Limit20:06
53m 48s
- Intro0:00
- Plug-in Procedure0:09
- Plug-in Procedure0:10
- Limit Laws9:14
- Limit Law 110:05
- Limit Law 210:54
- Limit Law 311:28
- Limit Law 411:54
- Limit Law 512:24
- Limit Law 613:14
- Limit Law 714:38
- Plug-in Procedure, Cont.16:35
- Plug-in Procedure, Cont.16:36
- Example I: Calculating Limits Mathematically20:50
- Example II: Calculating Limits Mathematically27:37
- Example III: Calculating Limits Mathematically31:42
- Example IV: Calculating Limits Mathematically35:36
- Example V: Calculating Limits Mathematically40:58
- Limits Theorem44:45
- Limits Theorem 144:46
- Limits Theorem 2: Squeeze Theorem46:34
- Example VI: Calculating Limits Mathematically49:26
21m 22s
- Intro0:00
- Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law0:16
- Example II: Evaluate the Following Limit1:51
- Example III: Evaluate the Following Limit3:36
- Example IV: Evaluate the Following Limit8:56
- Example V: Evaluate the Following Limit11:19
- Example VI: Calculating Limits Mathematically13:19
- Example VII: Calculating Limits Mathematically14:59
50m 1s
- Intro0:00
- Limit as x Goes to Infinity0:14
- Limit as x Goes to Infinity0:15
- Let's Look at f(x) = 1 / (x-3)1:04
- Summary9:34
- Example I: Calculating Limits as x Goes to Infinity12:16
- Example II: Calculating Limits as x Goes to Infinity21:22
- Example III: Calculating Limits as x Goes to Infinity24:10
- Example IV: Calculating Limits as x Goes to Infinity36:00
36m 31s
- Intro0:00
- Example I: Calculating Limits as x Goes to Infinity0:14
- Example II: Calculating Limits as x Goes to Infinity3:27
- Example III: Calculating Limits as x Goes to Infinity8:11
- Example IV: Calculating Limits as x Goes to Infinity14:20
- Example V: Calculating Limits as x Goes to Infinity20:07
- Example VI: Calculating Limits as x Goes to Infinity23:36
53m
- Intro0:00
- Definition of Continuity0:08
- Definition of Continuity0:09
- Example: Not Continuous3:52
- Example: Continuous4:58
- Example: Not Continuous5:52
- Procedure for Finding Continuity9:45
- Law of Continuity13:44
- Law of Continuity13:45
- Example I: Determining Continuity on a Graph15:55
- Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous17:57
- Example III: Is the Following Function Continuous at the Given Point?22:42
- Theorem for Composite Functions25:28
- Theorem for Composite Functions25:29
- Example IV: Is cos(x³ + ln x) Continuous at x=π/2?27:00
- Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?34:04
- Types of Discontinuity39:18
- Removable Discontinuity39:33
- Jump Discontinuity40:06
- Infinite Discontinuity40:32
- Intermediate Value Theorem40:58
- Intermediate Value Theorem: Hypothesis & Conclusion40:59
- Intermediate Value Theorem: Graphically43:40
- Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]47:46
40m 2s
- Intro0:00
- Derivative0:09
- Derivative0:10
- Example I: Find the Derivative of f(x)=x³2:20
- Notations for the Derivative7:32
- Notations for the Derivative7:33
- Derivative & Rate of Change11:14
- Recall the Rate of Change11:15
- Instantaneous Rate of Change17:04
- Graphing f(x) and f'(x)19:10
- Example II: Find the Derivative of x⁴ - x²24:00
- Example III: Find the Derivative of f(x)=√x30:51
53m 45s
- Intro0:00
- Example I: Find the Derivative of (2+x)/(3-x)0:18
- Derivatives II9:02
- f(x) is Differentiable if f'(x) Exists9:03
- Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other17:19
- Geometrically: Differentiability Means the Graph is Smooth18:44
- Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=020:53
- Example II: For x > 023:53
- Example II: For x < 025:36
- Example II: What is f(0) and What is the lim |x| as x→0?30:46
- Differentiability & Continuity34:22
- Differentiability & Continuity34:23
- How Can a Function Not be Differentiable at a Point?39:38
- How Can a Function Not be Differentiable at a Point?39:39
- Higher Derivatives41:58
- Higher Derivatives41:59
- Derivative Operator45:12
- Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³49:29
31m 38s
- Intro0:00
- Example I: Sketch f'(x)0:10
- Example II: Sketch f'(x)2:14
- Example III: Find the Derivative of the Following Function sing the Definition3:49
- Example IV: Determine f, f', and f'' on a Graph12:43
- Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value13:40
- Example VI: Distance vs. Time20:15
- Example VII: Displacement, Velocity, and Acceleration23:56
- Example VIII: Graph the Displacement Function28:20
47m 35s
- Intro0:00
- Differentiation of Polynomials & Exponential Functions0:15
- Derivative of a Function0:16
- Derivative of a Constant2:35
- Power Rule3:08
- If C is a Constant4:19
- Sum Rule5:22
- Exponential Functions6:26
- Example I: Differentiate7:45
- Example II: Differentiate12:38
- Example III: Differentiate15:13
- Example IV: Differentiate16:20
- Example V: Differentiate19:19
- Example VI: Find the Equation of the Tangent Line to a Function at a Given Point12:18
- Example VII: Find the First & Second Derivatives25:59
- Example VIII27:47
- Part A: Find the Velocity & Acceleration Functions as Functions of t27:48
- Part B: Find the Acceleration after 3 Seconds30:12
- Part C: Find the Acceleration when the Velocity is 030:53
- Part D: Graph the Position, Velocity, & Acceleration Graphs32:50
- Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents34:53
- Example X: Find a Point on a Graph42:31
47m 25s
- Intro0:00
- The Product, Power and Quotient Rules0:19
- Differentiate Functions0:20
- Product Rule5:30
- Quotient Rule9:15
- Power Rule10:00
- Example I: Product Rule13:48
- Example II: Quotient Rule16:13
- Example III: Power Rule18:28
- Example IV: Find dy/dx19:57
- Example V: Find dy/dx24:53
- Example VI: Find dy/dx28:38
- Example VII: Find an Equation for the Tangent to the Curve34:54
- Example VIII: Find d²y/dx²38:08
41m 8s
- Intro0:00
- Derivatives of the Trigonometric Functions0:09
- Let's Find the Derivative of f(x) = sin x0:10
- Important Limits to Know4:59
- d/dx (sin x)6:06
- d/dx (cos x)6:38
- d/dx (tan x)6:50
- d/dx (csc x)7:02
- d/dx (sec x)7:15
- d/dx (cot x)7:27
- Example I: Differentiate f(x) = x² - 4 cos x7:56
- Example II: Differentiate f(x) = x⁵ tan x9:04
- Example III: Differentiate f(x) = (cos x) / (3 + sin x)10:56
- Example IV: Differentiate f(x) = e^x / (tan x - sec x)14:06
- Example V: Differentiate f(x) = (csc x - 4) / (cot x)15:37
- Example VI: Find an Equation of the Tangent Line21:48
- Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?25:17
- Example VIII: Ladder Problem28:23
- Example IX: Evaluate33:22
- Example X: Evaluate36:38
24m 56s
- Intro0:00
- The Chain Rule0:13
- Recall the Composite Functions0:14
- Derivatives of Composite Functions1:34
- Example I: Identify f(x) and g(x) and Differentiate6:41
- Example II: Identify f(x) and g(x) and Differentiate9:47
- Example III: Differentiate11:03
- Example IV: Differentiate f(x) = -5 / (x² + 3)³12:15
- Example V: Differentiate f(x) = cos(x² + c²)14:35
- Example VI: Differentiate f(x) = cos⁴x +c²15:41
- Example VII: Differentiate17:03
- Example VIII: Differentiate f(x) = sin(tan x²)19:01
- Example IX: Differentiate f(x) = sin(tan² x)21:02
25m 32s
- Intro0:00
- Example I: Differentiate f(x) = sin(cos(tanx))0:38
- Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point2:25
- Example III: F(x) = f(g(x)), Find F' (6)4:22
- Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window5:35
- Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴10:18
- Example VI: Differentiate f(x) = sec²(12x)12:28
- Example VII: Differentiate14:41
- Example VIII: Differentiate19:25
- Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time21:13
52m 31s
- Intro0:00
- Implicit Differentiation0:09
- Implicit Differentiation0:10
- Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y12:15
- Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 1419:18
- Example III: Find (dy)/(dx) of x³y² + y³x² = 4x21:43
- Example IV: Find (dy)/(dx) of the Following Equation24:13
- Example V: Find (dy)/(dx) of 6sin x cos y = 129:00
- Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y31:02
- Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x37:36
- Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)41:03
- Example IX: Find (d²y)/(dx²) of x² + y² = 2544:05
- Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)47:48
47m 34s
- Intro0:00
- Linear Approximations & Differentials0:09
- Linear Approximations & Differentials0:10
- Example I: Linear Approximations & Differentials11:27
- Example II: Linear Approximations & Differentials20:19
- Differentials30:32
- Differentials30:33
- Example III: Linear Approximations & Differentials34:09
- Example IV: Linear Approximations & Differentials35:57
- Example V: Relative Error38:46
45m 33s
- Intro0:00
- Related Rates0:08
- Strategy for Solving Related Rates Problems #10:09
- Strategy for Solving Related Rates Problems #21:46
- Strategy for Solving Related Rates Problems #32:06
- Strategy for Solving Related Rates Problems #42:50
- Strategy for Solving Related Rates Problems #53:38
- Example I: Radius of a Balloon5:15
- Example II: Ladder12:52
- Example III: Water Tank19:08
- Example IV: Distance between Two Cars29:27
- Example V: Line-of-Sight36:20
37m 17s
- Intro0:00
- Example I: Shadow0:14
- Example II: Particle4:45
- Example III: Water Level10:28
- Example IV: Clock20:47
- Example V: Distance between a House and a Plane29:11
40m 44s
- Intro0:00
- Maximum & Minimum Values of a Function, Part 10:23
- Absolute Maximum2:20
- Absolute Minimum2:52
- Local Maximum3:38
- Local Minimum4:26
- Maximum & Minimum Values of a Function, Part 26:11
- Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min7:18
- Function with Local Max & Min but No Absolute Max & Min8:48
- Formal Definitions10:43
- Absolute Maximum11:18
- Absolute Minimum12:57
- Local Maximum14:37
- Local Minimum16:25
- Extreme Value Theorem18:08
- Theorem: f'(c) = 024:40
- Critical Number (Critical Value)26:14
- Procedure for Finding the Critical Values of f(x)28:32
- Example I: Find the Critical Values of f(x) x + sinx29:51
- Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]35:31
40m 44s
- Intro0:00
- Example I: Identify Absolute and Local Max & Min on the Following Graph0:11
- Example II: Sketch the Graph of a Continuous Function3:11
- Example III: Sketch the Following Graphs4:40
- Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²6:13
- Example V: Find the Critical Values of f(x) = |2x - 5|8:42
- Example VI: Find the Critical Values11:42
- Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]16:57
- Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]20:08
- Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]24:39
25m 54s
- Intro0:00
- Rolle's Theorem0:08
- Rolle's Theorem: If & Then0:09
- Rolle's Theorem: Geometrically2:06
- There May Be More than 1 c Such That f'( c ) = 03:30
- Example I: Rolle's Theorem4:58
- The Mean Value Theorem9:12
- The Mean Value Theorem: If & Then9:13
- The Mean Value Theorem: Geometrically11:07
- Example II: Mean Value Theorem13:43
- Example III: Mean Value Theorem21:19
25m 54s
- Intro0:00
- Using Derivatives to Graph Functions, Part I0:12
- Increasing/ Decreasing Test0:13
- Example I: Find the Intervals Over Which the Function is Increasing & Decreasing3:26
- Example II: Find the Local Maxima & Minima of the Function19:18
- Example III: Find the Local Maxima & Minima of the Function31:39
44m 58s
- Intro0:00
- Using Derivatives to Graph Functions, Part II0:13
- Concave Up & Concave Down0:14
- What Does This Mean in Terms of the Derivative?6:14
- Point of Inflection8:52
- Example I: Graph the Function13:18
- Example II: Function x⁴ - 5x²19:03
- Intervals of Increase & Decrease19:04
- Local Maxes and Mins25:01
- Intervals of Concavity & X-Values for the Points of Inflection29:18
- Intervals of Concavity & Y-Values for the Points of Inflection34:18
- Graphing the Function40:52
49m 19s
- Intro0:00
- Example I: Intervals, Local Maxes & Mins0:26
- Example II: Intervals, Local Maxes & Mins5:05
- Example III: Intervals, Local Maxes & Mins, and Inflection Points13:40
- Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity23:02
- Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity34:36
59m 1s
- Intro0:00
- Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes0:11
- Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes21:24
- Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D37:56
- Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes46:19
30m 9s
- Intro0:00
- L'Hospital's Rule0:19
- Indeterminate Forms0:20
- L'Hospital's Rule3:38
- Example I: Evaluate the Following Limit Using L'Hospital's Rule8:50
- Example II: Evaluate the Following Limit Using L'Hospital's Rule10:30
- Indeterminate Products11:54
- Indeterminate Products11:55
- Example III: L'Hospital's Rule & Indeterminate Products13:57
- Indeterminate Differences17:00
- Indeterminate Differences17:01
- Example IV: L'Hospital's Rule & Indeterminate Differences18:57
- Indeterminate Powers22:20
- Indeterminate Powers22:21
- Example V: L'Hospital's Rule & Indeterminate Powers25:13
38m 14s
- Intro0:00
- Example I: Evaluate the Following Limit0:17
- Example II: Evaluate the Following Limit2:45
- Example III: Evaluate the Following Limit6:54
- Example IV: Evaluate the Following Limit8:43
- Example V: Evaluate the Following Limit11:01
- Example VI: Evaluate the Following Limit14:48
- Example VII: Evaluate the Following Limit17:49
- Example VIII: Evaluate the Following Limit20:37
- Example IX: Evaluate the Following Limit25:16
- Example X: Evaluate the Following Limit32:44
49m 59s
- Intro0:00
- Example I: Find the Dimensions of the Box that Gives the Greatest Volume1:23
- Fundamentals of Optimization Problems18:08
- Fundamental #118:33
- Fundamental #219:09
- Fundamental #319:19
- Fundamental #420:59
- Fundamental #521:55
- Fundamental #623:44
- Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square24:36
- Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)35:13
- Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R43:10
55m 10s
- Intro0:00
- Example I: Optimization Problem0:13
- Example II: Optimization Problem17:34
- Example III: Optimization Problem35:06
- Example IV: Revenue, Cost, and Profit43:22
30m 22s
- Intro0:00
- Newton's Method0:45
- Newton's Method0:46
- Example I: Find x2 and x313:18
- Example II: Use Newton's Method to Approximate15:48
- Example III: Find the Root of the Following Equation to 6 Decimal Places19:57
- Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point23:11
55m 26s
- Intro0:00
- Antiderivatives0:23
- Definition of an Antiderivative0:24
- Antiderivative Theorem7:58
- Function & Antiderivative12:10
- x^n12:30
- 1/x13:00
- e^x13:08
- cos x13:18
- sin x14:01
- sec² x14:11
- secxtanx14:18
- 1/√(1-x²)14:26
- 1/(1+x²)14:36
- -1/√(1-x²)14:45
- Example I: Find the Most General Antiderivative for the Following Functions15:07
- Function 1: f(x) = x³ -6x² + 11x - 915:42
- Function 2: f(x) = 14√(x) - 27 4√x19:12
- Function 3: (fx) = cos x - 14 sinx20:53
- Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )22:10
- Function 5: f(x) = (3e^x) - 2/(1+x²)25:42
- Example II: Given the Following, Find the Original Function f(x)26:37
- Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 4027:55
- Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 530:34
- Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.232:54
- Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 737:54
- Example III: Falling Object41:58
- Problem 1: Find an Equation for the Height of the Ball after t Seconds42:48
- Problem 2: How Long Will It Take for the Ball to Strike the Ground?48:30
- Problem 3: What is the Velocity of the Ball as it Hits the Ground?49:52
- Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?50:46
51m 3s
- Intro0:00
- The Area Under a Curve0:13
- Approximate Using Rectangles0:14
- Let's Do This Again, Using 4 Different Rectangles9:40
- Approximate with Rectangles16:10
- Left Endpoint18:08
- Right Endpoint25:34
- Left Endpoint vs. Right Endpoint30:58
- Number of Rectangles34:08
- True Area37:36
- True Area37:37
- Sigma Notation & Limits43:32
- When You Have to Explicitly Solve Something47:56
33m 7s
- Intro0:00
- Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve0:10
- Example II: Using 5 Rectangles, Approximate the Area Under the Curve11:32
- Example III: Find the True Area by Evaluating the Limit Expression16:07
- Example IV: Find the True Area by Evaluating the Limit Expression24:52
43m 19s
- Intro0:00
- The Definite Integral0:08
- Definition to Find the Area of a Curve0:09
- Definition of the Definite Integral4:08
- Symbol for Definite Integral8:45
- Regions Below the x-axis15:18
- Associating Definite Integral to a Function19:38
- Integrable Function27:20
- Evaluating the Definite Integral29:26
- Evaluating the Definite Integral29:27
- Properties of the Definite Integral35:24
- Properties of the Definite Integral35:25
32m 14s
- Intro0:00
- Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals0:11
- Example II: Express the Following Limit as a Definite Integral5:28
- Example III: Evaluate the Following Definite Integral Using the Definition6:28
- Example IV: Evaluate the Following Integral Using the Definition17:06
- Example V: Evaluate the Following Definite Integral by Using Areas25:41
- Example VI: Definite Integral30:36
24m 17s
- Intro0:00
- The Fundamental Theorem of Calculus0:17
- Evaluating an Integral0:18
- Lim as x → ∞12:19
- Taking the Derivative14:06
- Differentiation & Integration are Inverse Processes15:04
- 1st Fundamental Theorem of Calculus20:08
- 1st Fundamental Theorem of Calculus20:09
- 2nd Fundamental Theorem of Calculus22:30
- 2nd Fundamental Theorem of Calculus22:31
25m 21s
- Intro0:00
- Example I: Find the Derivative of the Following Function0:17
- Example II: Find the Derivative of the Following Function1:40
- Example III: Find the Derivative of the Following Function2:32
- Example IV: Find the Derivative of the Following Function5:55
- Example V: Evaluate the Following Integral7:13
- Example VI: Evaluate the Following Integral9:46
- Example VII: Evaluate the Following Integral12:49
- Example VIII: Evaluate the Following Integral13:53
- Example IX: Evaluate the Following Graph15:24
- Local Maxs and Mins for g(x)15:25
- Where Does g(x) Achieve Its Absolute Max on [0,8]20:54
- On What Intervals is g(x) Concave Up/Down?22:20
- Sketch a Graph of g(x)24:34
34m 22s
- Intro0:00
- Example I: Evaluate the Following Indefinite Integral0:10
- Example II: Evaluate the Following Definite Integral0:59
- Example III: Evaluate the Following Integral2:59
- Example IV: Velocity Function7:46
- Part A: Net Displacement7:47
- Part B: Total Distance Travelled13:15
- Example V: Linear Density Function20:56
- Example VI: Acceleration Function25:10
- Part A: Velocity Function at Time t25:11
- Part B: Total Distance Travelled During the Time Interval28:38
27m 20s
- Intro0:00
- Table of Integrals0:35
- Example I: Evaluate the Following Indefinite Integral2:02
- Example II: Evaluate the Following Indefinite Integral7:27
- Example IIII: Evaluate the Following Indefinite Integral10:57
- Example IV: Evaluate the Following Indefinite Integral12:33
- Example V: Evaluate the Following14:28
- Example VI: Evaluate the Following16:00
- Example VII: Evaluate the Following19:01
- Example VIII: Evaluate the Following21:49
- Example IX: Evaluate the Following24:34
34m 56s
- Intro0:00
- Areas Between Two Curves: Function of x0:08
- Graph 1: Area Between f(x) & g(x)0:09
- Graph 2: Area Between f(x) & g(x)4:07
- Is It Possible to Write as a Single Integral?8:20
- Area Between the Curves on [a,b]9:24
- Absolute Value10:32
- Formula for Areas Between Two Curves: Top Function - Bottom Function17:03
- Areas Between Curves: Function of y17:49
- What if We are Given Functions of y?17:50
- Formula for Areas Between Two Curves: Right Function - Left Function21:48
- Finding a & b22:32
42m 55s
- Intro0:00
- Instructions for the Example Problems0:10
- Example I: y = 7x - x² and y=x0:37
- Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=26:25
- Example III: y=(1/x), y=(1/x³), and x=412:25
- Example IV: 15-2x² and y=x²-515:52
- Example V: x=(1/8)y³ and x=6-y²20:20
- Example VI: y=cos x, y=sin(2x), [0,π/2]24:34
- Example VII: y=2x², y=10x², 7x+2y=1029:51
- Example VIII: Velocity vs. Time33:23
- Part A: At 2.187 Minutes, Which care is Further Ahead?33:24
- Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?36:32
- Part C: At 4 Minutes Which Car is Ahead?37:11
- Part D: At What Time Will the Cars be Side by Side?37:50
34m 15s
- Intro0:00
- Volumes I: Slices0:18
- Rotate the Graph of y=√x about the x-axis0:19
- How can I use Integration to Find the Volume?3:16
- Slice the Solid Like a Loaf of Bread5:06
- Volumes Definition8:56
- Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation12:18
- Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation19:05
- Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation25:28
51m 43s
- Intro0:00
- Volumes II: Volumes by Washers0:11
- Rotating Region Bounded by y=x³ & y=x around the x-axis0:12
- Equation for Volumes by Washer11:14
- Process for Solving Volumes by Washer13:40
- Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis15:58
- Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis25:07
- Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis34:20
- Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis44:05
49m 36s
- Intro0:00
- Solids That Are Not Solids-of-Revolution0:11
- Cross-Section Area Review0:12
- Cross-Sections That Are Not Solids-of-Revolution7:36
- Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H10:54
- Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles20:39
- Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H29:27
- Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 6436:47
- Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis46:13
50m 2s
- Intro0:00
- Volumes by Cylindrical Shells0:11
- Find the Volume of the Following Region0:12
- Volumes by Cylindrical Shells: Integrating Along x14:12
- Volumes by Cylindrical Shells: Integrating Along y14:40
- Volumes by Cylindrical Shells Formulas16:22
- Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid18:33
- Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid25:57
- Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid31:38
- Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid38:44
- Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid44:03
32m 13s
- Intro0:00
- The Average Value of a Function0:07
- Average Value of f(x)0:08
- What if The Domain of f(x) is Not Finite?2:23
- Let's Calculate Average Value for f(x) = x² [2,5]4:46
- Mean Value Theorem for Integrate9:25
- Example I: Find the Average Value of the Given Function Over the Given Interval14:06
- Example II: Find the Average Value of the Given Function Over the Given Interval18:25
- Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]24:04
- Example IV: Find the Average Density of a Rod27:47
50m 32s
- Intro0:00
- Integration by Parts0:08
- The Product Rule for Differentiation0:09
- Integrating Both Sides Retains the Equality0:52
- Differential Notation2:24
- Example I: ∫ x cos x dx5:41
- Example II: ∫ x² sin(2x)dx12:01
- Example III: ∫ (e^x) cos x dx18:19
- Example IV: ∫ (sin^-1) (x) dx23:42
- Example V: ∫₁⁵ (lnx)² dx28:25
- Summary32:31
- Tabular Integration35:08
- Case 135:52
- Example: ∫x³sinx dx36:39
- Case 240:28
- Example: ∫e^(2x) sin 3x41:14
24m 50s
- Intro0:00
- Example I: ∫ sin³ (x) dx1:36
- Example II: ∫ cos⁵(x)sin²(x)dx4:36
- Example III: ∫ sin⁴(x)dx9:23
- Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx15:59
- #1: Power of sin is Odd16:00
- #2: Power of cos is Odd16:41
- #3: Powers of Both sin and cos are Odd16:55
- #4: Powers of Both sin and cos are Even17:10
- Example IV: ∫ tan⁴ (x) sec⁴ (x) dx17:34
- Example V: ∫ sec⁹(x) tan³(x) dx20:55
- Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx23:31
- #1: Power of sec is Odd23:32
- #2: Power of tan is Odd24:04
- #3: Powers of sec is Odd and/or Power of tan is Even24:18
22m 12s
- Intro0:00
- Trigonometric Integrals II0:09
- Recall: ∫tanx dx0:10
- Let's Find ∫secx dx3:23
- Example I: ∫ tan⁵ (x) dx6:23
- Example II: ∫ sec⁵ (x) dx11:41
- Summary: How to Deal with Integrals of Different Types19:04
- Identities to Deal with Integrals of Different Types19:05
- Example III: ∫cos(5x)sin(9x)dx19:57
17m 22s
- Intro0:00
- Example I: ∫sin²(x)cos⁷(x)dx0:14
- Example II: ∫x sin²(x) dx3:56
- Example III: ∫csc⁴ (x/5)dx8:39
- Example IV: ∫( (1-tan²x)/(sec²x) ) dx11:17
- Example V: ∫ 1 / (sinx-1) dx13:19
55m 12s
- Intro0:00
- Integration by Partial Fractions I0:11
- Recall the Idea of Finding a Common Denominator0:12
- Decomposing a Rational Function to Its Partial Fractions4:10
- 2 Types of Rational Function: Improper & Proper5:16
- Improper Rational Function7:26
- Improper Rational Function7:27
- Proper Rational Function11:16
- Proper Rational Function & Partial Fractions11:17
- Linear Factors14:04
- Irreducible Quadratic Factors15:02
- Case 1: G(x) is a Product of Distinct Linear Factors17:10
- Example I: Integration by Partial Fractions20:33
- Case 2: D(x) is a Product of Linear Factors40:58
- Example II: Integration by Partial Fractions44:41
42m 57s
- Intro0:00
- Case 3: D(x) Contains Irreducible Factors0:09
- Example I: Integration by Partial Fractions5:19
- Example II: Integration by Partial Fractions16:22
- Case 4: D(x) has Repeated Irreducible Quadratic Factors27:30
- Example III: Integration by Partial Fractions30:19
46m 37s
- Intro0:00
- Introduction to Differential Equations0:09
- Overview0:10
- Differential Equations Involving Derivatives of y(x)2:08
- Differential Equations Involving Derivatives of y(x) and Function of y(x)3:23
- Equations for an Unknown Number6:28
- What are These Differential Equations Saying?10:30
- Verifying that a Function is a Solution of the Differential Equation13:00
- Verifying that a Function is a Solution of the Differential Equation13:01
- Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation17:20
- General Solution22:00
- Particular Solution24:36
- Initial Value Problem27:42
- Example I: Verify that a Family of Functions is a Solution of the Differential Equation32:24
- Example II: For What Values of K Does the Function Satisfy the Differential Equation36:07
- Example III: Verify the Solution and Solve the Initial Value Problem39:47
28m 8s
- Intro0:00
- Separation of Variables0:28
- Separation of Variables0:29
- Example I: Solve the Following g Initial Value Problem8:29
- Example II: Solve the Following g Initial Value Problem13:46
- Example III: Find an Equation of the Curve18:48
51m 7s
- Intro0:00
- Standard Growth Model0:30
- Definition of the Standard/Natural Growth Model0:31
- Initial Conditions8:00
- The General Solution9:16
- Example I: Standard Growth Model10:45
- Logistic Growth Model18:33
- Logistic Growth Model18:34
- Solving the Initial Value Problem25:21
- What Happens When t → ∞36:42
- Example II: Solve the Following g Initial Value Problem41:50
- Relative Growth Rate46:56
- Relative Growth Rate46:57
- Relative Growth Rate Version for the Standard model49:04
24m 37s
- Intro0:00
- Slope Fields0:35
- Slope Fields0:36
- Graphing the Slope Fields, Part 111:12
- Graphing the Slope Fields, Part 215:37
- Graphing the Slope Fields, Part 317:25
- Steps to Solving Slope Field Problems20:24
- Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y22:38
45m 29s
- Intro0:00
- Exam Link0:10
- Problem #11:26
- Problem #22:52
- Problem #34:42
- Problem #47:03
- Problem #510:01
- Problem #613:49
- Problem #715:16
- Problem #819:06
- Problem #923:10
- Problem #1028:10
- Problem #1131:30
- Problem #1233:53
- Problem #1337:45
- Problem #1441:17
41m 55s
- Intro0:00
- Problem #150:22
- Problem #163:10
- Problem #175:30
- Problem #188:03
- Problem #199:53
- Problem #2014:51
- Problem #2117:30
- Problem #2222:12
- Problem #2325:48
- Problem #2429:57
- Problem #2533:35
- Problem #2635:57
- Problem #2737:57
- Problem #2840:04
58m 47s
- Intro0:00
- Problem #11:22
- Problem #24:55
- Problem #310:49
- Problem #413:05
- Problem #514:54
- Problem #617:25
- Problem #718:39
- Problem #820:27
- Problem #926:48
- Problem #1028:23
- Problem #1134:03
- Problem #1236:25
- Problem #1339:52
- Problem #1443:12
- Problem #1547:18
- Problem #1650:41
- Problem #1756:38
25m 40s
- Intro0:00
- Problem #1: Part A1:14
- Problem #1: Part B4:46
- Problem #1: Part C8:00
- Problem #2: Part A12:24
- Problem #2: Part B16:51
- Problem #2: Part C17:17
- Problem #3: Part A18:16
- Problem #3: Part B19:54
- Problem #3: Part C21:44
- Problem #3: Part D22:57
31m 20s
- Intro0:00
- Problem #4: Part A1:35
- Problem #4: Part B5:54
- Problem #4: Part C8:50
- Problem #4: Part D9:40
- Problem #5: Part A11:26
- Problem #5: Part B13:11
- Problem #5: Part C15:07
- Problem #5: Part D19:57
- Problem #6: Part A22:01
- Problem #6: Part B25:34
- Problem #6: Part C28:54
0 answers
Post by Leonardo Luo on April 14, 2021
My "post comment" button is tilted to the side...
1 answer
Last reply by: Professor Hovasapian
Thu Oct 19, 2017 5:39 AM
Post by Maya Balaji on October 18, 2017
For example IV- is there not supposed to be an x in the numerator along with the 30 for the answer? I'm just not sure where the x that corresponded with 2 (2x) went. Thanks so much!
1 answer
Last reply by: Professor Hovasapian
Wed Jan 18, 2017 6:17 PM
Post by Sarmad Khokhar on December 26, 2016
Every Calculus teacher should teach chain rule through this method
1 answer
Last reply by: Professor Hovasapian
Fri Nov 18, 2016 8:18 PM
Post by Muhammad Ziad on November 9, 2016
Hello Professor Hovasapian,
In example 10, why wasn't the chain rule also applied to (10x-2)? Wouldn't it then be:
8(5x^2-2x-2)^7 . (10x-2) . (10) . (x^2-2)^4
Thank you so much for this lecture, I found it very helpful!
1 answer
Last reply by: Professor Hovasapian
Wed Oct 26, 2016 7:15 PM
Post by Kitt Parker on October 21, 2016
Professor,
I believe we dropped the X off of 2x when we tied everything together in example IV. Shouldn't we have 30x/(x^2+3)^4? If that is not the case, what dissolved the x from 2x? Thank you for the extra guidance.
1 answer
Last reply by: Professor Hovasapian
Sun Sep 4, 2016 5:41 AM
Post by Isaac Martinez on September 2, 2016
OMG!!! I finally understand the chain rule and the quotient rule!!! Major breakthrough in my life!!! Thank you Professor Hovasapian!!!!
1 answer
Last reply by: Professor Hovasapian
Fri Apr 1, 2016 2:53 AM
Post by Venise Martinez on March 30, 2016
AWESOME SAUCE!
1 answer
Last reply by: Professor Hovasapian
Fri Mar 25, 2016 10:57 PM
Post by john lee on March 10, 2016
Professor Hovasapian,
I found that you skip the derivatives of inverse trigonometric functions. My 5 steps to a 5 is 2013 edition. Is it deleted in 2015 edition or it is still important for AP exam?
thanks for help!
1 answer
Last reply by: Professor Hovasapian
Thu Feb 11, 2016 10:59 PM
Post by Zekarias Araya on February 11, 2016
Perfect!!