AP Calculus AB Derivatives of the Trigonometric Functions

Hello, welcome back to www.educator.com and welcome back to AP Calculus.0000

Today, we are going to be talking about the derivatives of the trigonometric functions.0004

Let us jump right on in.0008

Let us start off by finding the derivative of sin x.0010

Let me see, I will stick with blue today.0016

Let us find the derivative of f(x) = the sin(x).0024

We have the derivative definition.0040

It is going to be the limit, as h approaches 0 of f(x + h - f(x)/ h).0045

We are going to form quotient and I take the limit as h goes to 0.0055

I’m just going to put little quotation marks here for the limit h approaches 0.0061

I do not want to keep writing it, I hope you will forgive.0064

Our function is sin(x).0068

We have sin(x) + h – sin(x)/ h.0070

And then, when we expand this sin(x) + h, you remember the addition formulas0080

or you remember having seen the addition formulas, it becomes sin x cos h + cos x sin h + sin x, we have all of that /h.0086

What I’m going to do is I’m going to separate these out.0104

I’m going to combine a couple of these.0108

Let me go ahead and put a.0111

This is the limit as h approaches 0 of, I’m making myself a little more room here.0112

We are going to have sin x cos h – sin x/ h + this second part which is cos x sin h/ h.0120

This is going to be, this is going to equal that.0144

This is going to equal the limit as h approaches 0.0147

Let me see, I’m going to take out the sin x.0152

The sin x × this is going to be cos h -1/ h.0156

I’m keeping the x and h together.0166

That is all I’m doing here.0168

+ cos x × sin h/ h.0170

That is it, I just put things together so that I can actually do it.0177

Let me go ahead and write it here.0183

In this case, this is the limit as h approaches 0.0184

This × this + this × this.0188

It is going to be the limit of this × the limit of this + the limit of this × the limit of this.0191

Limit theorems, let me just go ahead and take them one at a time.0198

This actually equal the limit as h approaches 0 of sin x × the limit as h approaches 0 of cos h -1/ h0202

+ the limit as h approaches 0 of cos x × the limit as h approaches 0 of sin h/ h.0220

Here is what happens.0234

As h goes to 0, notice there is no h in here.0235

The limit as h approaches 0 of sin x is just sin x.0238

This is just sin x.0244

This limit right here, let me do this in red.0248

These two limits, this limit and this limit are very important limits.0253

I’m not going to demonstrate how they end up being what they equal but these are two limits that you definitely have to know.0258

This one is going to be 0.0266

It is sin x × 0 + the limit as h goes to 0 of cos x is just cos x.0268

This limit right here, the limit of the sin of h/ h as h goes to 0 is actually equal to 1.0275

Let us see, therefore, the derivative of sin x = cos x, this term goes to 0.0285

Once again, these two limits are very important to know.0299

You just have to know them, very important to know.0309

First one, the limit as x approaches 0 of the sin(x)/ x that is equal to 1.0320

The other limit, the limit as x approaches 0 of cos x -/ x that is equal to 0.0332

We will be using these limits to evaluate other limits, until the time that we learn something called L’Hospitals rule.0348

And then, we would not have to worry about these limits anymore.0355

But again, they are very important limits, they do tend to come up.0357

As far as the formulas are concerned, let us go back to black here.0361

The derivative with respect to x of the sin(x) = the cos of x.0371

Going with the same process of the different quotient, taking the limit.0378

Again, we will not go through the pool for each one, we are just going to list them because again,0381

what we want to do is we just want to develop technique.0385

We want to be able to differentiate quickly.0390

In this part of the course, this is all we are doing, finding ways to differentiate all this different functions.0392

The derivative with respect to x of cos x is equal to -sin x.0399

Make sure you watch your sign.0406

Derivative of sin is cos, derivative of cos is –sin.0407

The derivative of the tangent function, the tan of x = sec² x.0414

The derivative of cosec of x = -cosec x cot x.0424

The derivative with respect to x of sec x is equal to sec x tan x.0436

The derivative of the cotangent function, cot x is equal to –cosec² x.0448

These are our 6 trigonometric functions and their derivatives.0456

The rest is now just examples.0460

Let us jump right onto the examples and see what we can do.0464

Again, really straightforward, nothing particularly complicated.0467

Just a question of memorizing via derivative formulas and doing a bunch of problems.0471

Differentiate f(x) = x² – cos(x), very simple.0478

F’(x), if you want you can use the dy dx symbolism as well, it does not really matter.0484

You are going to see both.0496

The derivative of this is 2x 4 cos x, it is going to be -, this – stays and this is going to be -4 sin x.0500

The 4 is just a constant.0513

Let us go ahead and take the constant to be consistent.0516

-4 × the derivative of cos x which is –sin x.0520

Our f’(x) is equal to 2x, - and -, +4 sin x.0528

There you go, nice and straightforward.0536

You just have to use the differentiation formulas.0538

All we are doing is practice, practice, practice.0541

Example 2, differentiate f(x) = x⁵ tan x.0546

Here we have a product rule.0551

This is for our first function, this is our second function.0553

We have product rule, in this case one of them is a monomial and the other one is a trigonometric function.0556

It is going to be the derivative of this × that + the derivative of this × that.0563

F’(x) is equal to the derivative of this is 5x⁴ × this × the tan(x) + the derivative of this × that + sec² x × x⁵.0569

We just want to go ahead and bring this x⁵ in front.0593

You do not have to, you can leave it like this, it is not a problem.0596

Traditionally, we tend to bring everything forward and leave the trigonometric function at the end.0599

We have 5x⁴ tan(x) + x⁵ sec² x.0604

That is it, nice and simple.0614

Do you want to simplify it more, do you want to express it in terms of sin and cos, you can if you want to.0618

Simplification is really going to be, it is a totally individual thing how far you want to simplify.0623

You remember from product rule, quotient rule, in the previous lessons.0629

Where do you stop, at some point, you just have to stop.0634

Ultimately, it does not really matter where you stop.0638

You want to simplify as much as reasonable but it is going to be subjective.0640

What is reasonable to you might not be reasonable to somebody else.0645

Ultimately, you just have to ask what is reasonable to your teacher.0648

That takes care of that one.0653

Differentiate x(x) = cos x/3 + sin x.0657

Now we have a quotient rule and some trigonometric functions.0661

You remember the quotient rule, if you have f/g and if you take the derivative, you are going to have g f’ – f g’/ g².0665

This is f and this is g, f is cos x, 3 + sin x is g.0680

Therefore, we have f’(x) is equal to this × the derivative of that.0685

We have 3 + sin x × the derivative of cos x which is –sin x, - this cos x × the derivative of this.0691

The derivative of this is 0 +, this is just cos x/ 3 + sin x².0705

Let us go ahead and distribute this out.0719

Let us see.0724

You know what, I think I made a mistake on my paper so I will do it here, just as is.0732

Sin x cancels, this is going to be -3 sin x – sin² x – cos² x/ 3 + sin x².0736

Let me see, I have got -3 sin x.0762

I’m going to write this part -.0767

I’m going to write this as sin² x + cos² x.0772

The – distributes giving me the two negatives, /3 + sin x².0778

This right here is the identity.0790

This is just 1.0793

Our final answer -3 sin x -1/ 3 + sin x².0797

Let us go ahead and factor out that negative.0814

Let us write it as -3 sin x + 1.0819

No, that is not going to simplify anything.0830

I thought maybe that something might cancel but it is not.0832

You know what, we can stop there, not a problem.0835

There you go, quotient rule, trigonometric functions.0839

Reasonable, very straightforward.0843

Now we have differentiate e ⁺x tan x – sec x.0848

For these two, we have product rule.0855

We have e ⁺x and tan x – sec x, we just differentiate straight.0857

F’(x) = this × the derivative of that = e ⁺x × sec² x + the derivative of this × that.0862

I will leave it in the front.0877

It is the derivative of this × that + the derivative of,0883

Wait a minute, what am I doing here, let us start again.0889

The derivative of this × that.0895

We have e ⁺x × tan x, there we go, + the derivative of this × that +, I will leave this in front, e ⁺x sec² x.0896

And then, -the derivative of the sec x which is sec x tan x.0912

That is fine, you can go ahead and leave it like that.0923

You have a tan x tan x, sec² x sec² x, e ⁺x.0926

You might as well leave it like that, I do not think it will make it any easier.0931

That was nice and simple, good.0935

Again, we have another quotient rule.0940

Let us see what we can do with this one.0943

F'(x) is equal to this × the denominator × the derivative of the numerator.0947

We have cot(x) × the derivative of this which is going to be -cosec x cot x – this0954

which is cosec x – 4 × the derivative of the cot x which is –cosec² x/ the cot² (x).0971

If you want you can write this cot(x)², not a problem.0990

It is just we tend to put these squared on the cot.0996

Whatever is easier for you, you will see in a little bit that I actually do write that sometimes.0999

In the next lesson, when we talk about something called the chain rule, it is often easier.1006

At least in the beginning, when you are getting used to it.1011

To write something like the cos² (x), it is actually easier to write it as cos(x)².1013

It tends to make the differentiation a little bit easier.1021

Again, as you sort of get accustomed to the idea of seeing the squared here.1023

Let us see what we can do with this.1029

Let us go ahead and multiply this out, I guess the numerator.1030

We have –cosec x cot² x.1037

This is – × -, we have + cosec³ x.1046

This is – × - × -, it stays -4 cosec² x/ cot² x.1056

cosec and cosec, I’m going to pull out a cosec.1072

-cosec x × cot² x – cosec² x.1074

The – and – gives me the +.1098

And then, this is going to be a -, so it is going to be a +.1101

-cosec, that is going to be -, it is going to be +.1110

This is -, that is -, this is going to be +4 cosec² x.1114

It is always not that easy to keep track of all these signs and things.1122

By all means, if I make a mistake, it is definitely going to happen.1125

That is the nature of the game.1129

You have all these symbols floating around.1131

Mistakes are going to happen.1133

/cot² x.1135

Let us see, I think I got this right.1141

Let me double check.1143

-cosec, I pulled out a negative.1144

This becomes -, - × - is + goes to +.1147

This is -, so -.1151

Yes, I think we are good.1152

This becomes, I’m going to write this –cosec as -1/ sin(x).1155

Here, cot² – cosec², if you remember from your trigonometric identities, it is equal to 1.1162

This is going to be 1 + 4 cosec² x.1169

The cot² x, I’m going to write as cos² x/ sin² x.1177

When I do that, I’m going to go ahead and cancel this sign and one of these signs.1186

Let us see, that is going to give me.1193

Now I have f’(x) is equal to,1201

Let me write that part over, actually.1209

-1/ sin(x) × 1 + 4 cosec(x).1211

I think that was what is on top.1220

Here we have cos² x/ sin² x.1222

We cancel the sin and we can flip this.1226

This sin ends up coming to the top as sin x, the – stays.1230

Here we have 1 + 4 cosec x.1237

Over here, we have cos² x stays on the bottom.1243

Then, let me write this as sin(x) × 1 + 4/ cosec is sin(x)/ cos² (x).1254

Again, you do not really need to simplify all this much.1275

But to solve the identities, let us take it as far as we can take it.1277

This is going to be –sin(x), when I distribute this.1284

This, the sin x are going to cancel so it is going to be -4/ cos² (x).1289

I think that should take care of it.1298

I think that is as far as we want to take that one.1303

There you go.1307

Find an equation of the line tangent to y = x² sin² x at x = π/6.1310

We are looking for an equation of the tangent line.1318

We are definitely going to be using y – y1 = m × x – x1.1321

We have x, that is π/6, it is going to be our x1.1331

Let us go ahead and find the y1 first.1335

Y(π/6) is going to equal what we just put in there.1339

It is going to be π/6² × sin(π/6).1346

π/6² is π²/36 × sin(π/6), sin(30 degrees) is 0.5 or ½.1352

We end up with π²/72.1363

We know our point.1370

Now, π/6 and π²/72, that is tan.1372

Tan which means find the derivative.1383

Y’, this is a product rule.1386

It is going to be the derivative of this × this.1391

We get 2x × sin(x) + the derivative of this × this + cos(x) × x².1393

Y’ at π/6, we are trying to find the slope.1406

It is going to equal to × π/6 × sin(π/6) + cos(π/6) × π/6².1414

What do we have here?1433

Let us just go ahead and write it all out, it is not a problem.1436

This is going to be, let me do it over here.1440

2 × π/6, sin(π/6) is ½, cos(π/6) is √3/2, × π²/ 36.1443

Here we have, the 2 and 2 cancels.1458

We are left with π/6 + π² √3/72.1460

This is going to be our slope.1473

That is our slope.1477

Now we put it into the equation.1481

We get y – y1 which is π²/72 = the slope which is π/6 + π² √3/72 × x – π/6.1484

There you go, that is your equation.1505

Again, depending on the extent to which your teacher wants you to simplify that, you can go ahead and do so.1509

Nice, nothing particularly strange.1515

For what values of x does the graph of the function x + 3 cos(x) have a horizontal tangent?1520

Horizontal tangent means that the derivative is 0.1527

Horizontal tangent that implies that f’(x) is equal to 0.1536

Let us go ahead and take the derivative of this.1544

If f(x) = x + 3 × cos(x), that means that f’(x) is going to equal 1.1548

The derivative of cos(x) is – sin x, we have -3 sin x.1560

We are going to set that equal to 0.1566

1 – 3 sin x = 0, that means that 3 sin(x) is equal to 1.1569

That means sin(x) is equal to 1/3.1577

Therefore, x is equal to the inv sin(1/3).1581

This is a positive number.1588

The sin is positive in two quadrants.1592

We are going to get two answers.1594

We are going to get an angle of this quadrant.1595

We are going to get an angle of this quadrant.1597

The angle that we are taking, the first one is going to be this one.1599

The second one is going to be that one.1602

When I do this, when I put this into the calculator, I get x = in radians, 0.3398 radians.1607

That is this first angle.1619

The other angle, that is going to be 0.3398, that is going to be this angle over here but we do not measure it from here.1623

We take all of our measurements from the +x axis.1630

We are going to take π which is 180 - 0.3398.1632

π - 0.3398 rad.1638

There you go, once again, we draw this out.1650

This angle is x1, the 0.3398 it happens to be 19.47°.1653

The other one s 19.47°, in the second quadrant but we measured it from here.1666

This is x2, this is the π - 0.3398.1675

It happens to equal what is going to be 180 - 19.47.1680

We have got 160.52°.1684

There you, these are our two answers.1689

Of course, every 2π after that, it comes around again.1693

It is a periodic function.1699

Let us see, word problem, nice.1704

A ladder 12ft long rest against the wall, if the bottom of the ladder slides from the wall,1710

how fast does the height of the ladder against the wall change with respect to the angle that the ladder makes with the wall,1715

as the ladder is being pulled away at the base?1723

Let us go ahead and draw a picture here.1726

We have a wall here and we have our ladder.1729

Let me just go ahead.1736

We have our ladder that way.1740

They want to know, this ladder is 12ft long.1742

The bottom of the ladder slides away.1747

Now, this ladder is going to be moving this way which means that the top of the ladder is going to be moving that way.1748

How fast does the height of the ladder against the wall change?1755

How fast does this height changing?1758

Let us call this height y, let us call this x.1761

With respect to the angle that the ladder makes with the wall.1766

The angle that the ladder makes with the wall, let us call it θ.1769

They want to know how fast does the height of the ladder against the wall change, with respect to the angle that the ladder makes.1776

They want to know dy dθ.1782

They want to know how fast this changes, y with respect to the change in the angle θ.1788

That is dy dθ, that is it.1797

What that means is we need to find an equation y = some function of θ and we need to differentiate it, y’.1802

Y’ is dy dθ.1813

If this is what you are looking for, you need to find an equation of this equal to some function of this.1819

And then, you differentiate it and that will give you what this is.1825

You have a relationship here between y and θ and 12.1829

We have that the cos(θ) is equal to 1/12 which means that y = 12 × cos(θ).1834

Y’ which is equal to dy dθ is equal to -12 × cos(θ).1850

We are done, we have what we wanted.1860

The rate of change of y is equal to -12 × cos of whatever θ happens to be at that moment.1865

If θ happens to be some number that means y is changing.1874

Notice, it is negative, it s getting smaller.1880

The ladder is falling, it is going this way.1884

This negative sign confirms the fact that it is getting smaller.1886

If I were to do dx dθ, I would find that this is actually positive because it is getting bigger.1891

All of this is confirmed.1897

Once again, we are looking for dy dθ.1899

How fast does y change, when I change θ?1902

That means I have to find an equation of this, in terms of this, y as a function of θ.1905

I found it, now I just differentiated it and I have what I want.1911

Nice and simple.1915

If they gave us a specific θ, you would put it in and solve.1916

For example, if they said π/3.1920

Cos(π/3) is ½ so -12 × ½, it is going to be -6.1924

The rate of change of y when θ actually hits π/3, it is going to be changing by 6ft/s, something like that.1932

There you go, very important set of problems, this whole idea of the rate of change.1941

That is what we are doing, the rate of change.1951

These word problems in calculus are going to be profoundly important.1952

Again, the difficult part is not the actual differentiation, that is the easiest part of the whole problem.1958

The difficulty with these word problems is going to be taking a look at this and constructing what is going on,1963

and extracting an equation among the variables that you are interested in.1968

Keeping track of what it is that you want.1974

Very important.1976

When they say how fast does the height of the ladder against the wall change with respect to the angle?1978

That is dh dθ or in this case I chose y as my height, dy dθ.1985

I need an equation y, in terms of θ, that is just automatic.1991

That is what is going on, this is what you want to extract from these word problems.1995

Let us take a look at example number 9.2003

Evaluate the limit as x approaches 0 of tan(7x)/x.2005

Again, now we are going to use those limits that we saw.2012

The limit of sin x/ x and limit of cos x – 1/x.2015

The limit as x goes to 0.2020

We are going to use them to evaluate these limits.2021

Let us see what we can do.2025

In this particular case, we have 7x/ x.2027

This is not tan(x), this not sin(x)/x.2029

We need to sort of fiddle with this.2034

Let us see what we can do.2037

Let us see if we can somehow transform this and turn it into something that looks like something we already know.2039

What I'm going to do, I’m going to work with tan(7x)/x.2047

I’m going to leave the limit off for the time being.2052

I will come back to it at the end, when I simplify it.2055

I’m going to multiply this by 7/7.2058

I’m just multiplying by 1.2060

I’m not actually changing anything.2061

This equals 7 × tan(7x)/ 7x.2064

This is nice, now I have the argument of the trigonometric function and the denominator the same.2072

It starts to look interesting.2077

I’m going to express tan(7x), this is 7.2079

The tan(7x) is equal to sin(7x)/ cos(7x).2088

I’m going to write this as 7 × sin(7x)/ 7x × cos(7x).2097

That is it, all I have done is change the tangent to this.2107

I’m going to take this and this, and then this, separately.2110

This is equal to 7 × sin(7x)/ 7x × 1/ cos(7x).2120

Now I’m going to take the limit of that.2133

The limit as x approaches 0 of this thing 7 × sin(7x)/ 7x × 1/ cos(7x).2136

The limit as x approaches 0 of 7 is just 7.2150

The limit as x approaches 0 of sin(7x)/7x that is where we use that limit that we have.2153

The limit as x approaches 0 of sin(a)/a is equal to 1.2160

Here, as x approaches 0, cos(7) × 0 is cos(0).2168

Cos(0) is 1, 1/1 is 1.2173

Our final answer is 7.2176

I have transformed this thing, tan(7x)/ x by just manipulation into something that I actually recognize, using what is able to find that.2178

This limit is easy, this limit is easy, I get 7.2191

I hope that make sense.2195

Evaluate the limit as x approaches infinity, sin(1)/ x.2200

Let us take this x × sin(1)/ x.2208

We have x × sin(1)/ x.2220

I’m going to multiply by 1, in the form of 1/x/ 1/x.2224

That is equal to 1/x × x × sin(1)/x / 1/x.2233

I will just multiply 1/x and 1/x down below.2248

This and this cancel.2253

I’m left with the sin(1)/x/ 1/x.2254

I have this thing.2261

Now we take the limit, but this time the limit is saying as x goes to infinity of this thing.2269

I do not like doing that.2278

Sin(1)/x/ 1/x, I like it to be vertical.2279

This and this = a.2286

As x goes to infinity, as this goes to infinity, this whole thing goes to 0.2293

A is going to 0.2301

This is the same as the limit as x goes to 0 of sin(a)/a.2304

We know what this limit is already, it is equal to 1.2317

That is it, mathematical manipulation.2320

The question that most kids ask is that how do you know how to do this?2326

You do not, there is no way of looking at a problem and knowing exactly what steps you are going to take.2329

You just try.2335

What you saw here is the finished product.2337

What you did not see was the 5 or 6 different things that we try to do,2341

manipulations that did not go anywhere, that we ended up hitting a wall.2346

That is what is going to happen.2350

What you want to do in calculus from this point forward in your math work and your science work,2352

you want to disabuse yourself of this notion that you are supposed to look at a problem2359

and just all of the sudden know exactly what to do.2364

There are too many steps at this point in higher math and higher science,2366

for you to actually be able to see the entire path in your head.2371

Sometimes you will but sometimes you just have to start.2374

Especially, when it involves some form of mathematical manipulation to make the problem a little bit more practical.2377

You just have to start and you see where you go.2382

Maybe you would have tried x/x, that did not do anything.2384

You just try and if that does not work, if you hit a wall, you go back and you start again.2388

In this case, it turned out to be 1/x and 1/x.2393

It fell out, it fell out, perfect.2396

You get your answer that way.2398

Trust the mathematics.2400

You want to trust the mathematics and you want to trust your intuition.2404

But do not trust the fact that just because you do not see the entire path, that you know how to solve the problem.2408

You need to be able to work a couple of steps at a time because the solution might be so far ahead that you cannot actually see it.2415

The solution might be, there might be a curve, it might be along the curve.2422

You cannot see the curve ahead but if you take a couple of steps toward it, then you can see the curve ahead.2426

That is how this works.2431

Just do that, do not worry about it.2434

It is not like we look at these problems and automatically know what to do.2437

Most of the time, we do not, particularly with the things that we deal within our research or anything else.2441

We do not know, when we try things, most of the time we actually fail.2447

What you are seeing is the actual define of success.2452

It makes it look like this is really simple, let us just do this, no.2456

Anyway, I hope that helps.2461

Take good care and I will see you next time.2464

Thank you for joining us here at www.educator.com, bye.2466