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College Calculus: Level I The Chain Rule

Section 3: Derivatives, part 1: Lecture 7 | 23:47 min

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Lecture Comments (9)

	0 answers Post by Rafael Mojica on January 28, 2015 Where did you get the 6 from? at the last example in the final answer you multiplied by 6.
	0 answers Post by Josh Winfield on December 17, 2012 using two variables on example three is so much simpler i found. f(u)=u^2 g(x)=6sin(6x)
	0 answers Post by Stephanie DahlstrÃ¶m on December 11, 2012 I wonder how it works when you have a function like this: f(x) = t/(t^2+1)^0,5 I've tried the quotient rule and the chain rule but I don't get the right answer.
	0 answers Post by Gary Becraft on October 19, 2011 that was much easier to understand than the explanation given in class. I have heard others use the phrase "the stuff" and it simplifies the concept nicely.
	0 answers Post by Atticus McCoy on October 12, 2011 Awesome job I'm loving it :)
	3 answers Last reply by: StefÃ¡n Berg Jansson Sat Nov 26, 2011 6:31 PM Post by Abel Morales on February 19, 2011 I have a question, sin^2 (6x) wouldn't that equal cos^2 (6x), if not, why?

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The Chain Rule

Think of the function you are trying to differentiate as a function composition.
You can informally think of this as involving an “outer function” and an “inner function.”
Some problems involve a composition of three or more functions!
You will be using the Chain Rule throughout the rest of calculus, so it is important to learn it really well!

The Chain Rule

Find f′(x) if f(x) = (3x⁵ + 7x + 11)¹³

u(x) = 3x⁵ + 7x + 11
[du/dx] = u′(x) = 15x⁴ + 7
f(x) = u¹³
f′(x) = [d/dx] u¹³
f′(x) = 13 u¹² [du/dx]

13(3x⁵ + 7x + 11)¹² (15x⁴ + 7)

Find f′(x) if f(x) = √{x³ + 5x}

u = x³ + 5x
[du/dx] = 3x² + 5
f′(x) = [d/dx] √u
f′(x) = [d/dx] u^[1/2]
= [1/2] (u^−[1/2]) [du/dx]
= [du/(2√u)] =

[(3x² + 5)/(2√{x³ + 5x})]

Find f′(x) if f(x) = sin²(x)

u = sin(x)
u′ = cos(x)
f′(x) = [d/dx] u²
= 2 u u′
= 2 sin(x) cos(x) =

sin(2x)

Find f′(x) if f(x) = (sec(5x))⁹

u = sec(5x), v = 5x, v′ = 5
u = sec(v)
u′ = sec(v)tan(v) v′
u′ = 5sec(5x)tan(5x)
f′(x) = [d/dx] u⁹
= 9 u⁸ u′
= 9 (sec(5x))⁸ (5 sec(5x) tan(5x)) =

45 tan(5x) (sec(5x))⁹

Find f′(x) if f(x) = cot(5x² + 3x)

f′(x) = −csc²(5x² + 3x) [d/dx] (5x² + 3x)
= −csc²(5x² + 3x) (10x + 3) =

−(10x + 3)csc²(5x² + 3x)

Find f′(x) if f(x) = [(x²)/tan(3x)]

g = x², g′ = 2x
h = tan(3x), h′ = sec²(3x) [d/dx] 3x = 3 sec²(3x)
f′(x) = [d/dx] [g/h]
= [(h g′− g h′)/(h²)]
= [(tan(3x) (2x) − x² (3 sec²(3x)))/(tan²(3x))] =

[(2x tan(3x) − 3x² sec²(3x))/(tan²(3x))]

Find f′(x) if f(x) = (x² + 8x + 16)(x + 4)³

f(x) = (x² + 8x + 16) (x + 4)³
= (x + 4)² (x + 4)³
= (x + 4)⁵
f′(x) = 5 (x + 4)⁴ [d/dx] (x + 4)

5 (x + 4)⁴

Find f′(x) if f(x) = (7x⁹ + √{3x})(x + 1)⁴

u = 7x⁹ + √{3x}, u′ = 63x⁸ + [(√3)/(2√x)]
v = (x + 1)⁴, v′ = 4 (x + 1)³ [d/dx] (x + 1) = 4 (x + 1)³
f′(x) = [d/dx] u v
= u v′+ v u′=

(7x⁹ + √{3x})(4 (x + 1)³) + (x + 1)⁴ (63x⁸ + [(√3)/(2√x)])

Find f′(x) if f(x) = sin(sin(x))

u = sin(x), u′ = cos(x)
f′(x) = [d/dx] sin(u)
= cos(u) u′=

cos(sin(x)) cos(x)

Find f′(x) if f(x) = √{csc(11x)}

u = csc(11x), v = 11x, v′ = 11
u = csc(v), u′ = −cot(v)csc(v) v′
u′ = −11 cot(11x)csc(11x)
f′(x) = [d/dx] u^[1/2]
= [1/(2√u)] u′
= [(−11 cot(11x)csc(11x))/(2√{csc(11x)})] =

[(−11cot(11x)√{csc(11x)})/2]

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Answer

The Chain Rule

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Intro

Statement of the Chain Rule

Chain Rule for Three Functions

Lecture Example 1

Lecture Example 2

Lecture Example 3

Additional Example 4

Additional Example 5

Intro 0:00
Statement of the Chain Rule 0:09

Chain Rule for Three Functions

Lecture Example 1 1:00
Lecture Example 2 4:34
Lecture Example 3 7:23
Additional Example 4
Additional Example 5

College Calculus 1 Online Course

Section 1: Overview of Functions
	Review of Functions	26:29
	Compositions of Functions	12:29
Section 2: Limits
	Average and Instantaneous Rates of Change	20:59
	Limit Investigations	22:37
	Algebraic Evaluation of Limits	28:19
	Formal Definition of a Limit	23:39
	Continuity and the Intermediate Value Theorem	19:09
Section 3: Derivatives, part 1
	Limit Definition of the Derivative	22:52
	The Power Rule	26:01
	The Product Rule	14:54
	The Quotient Rule	19:17
	Applications of Rates of Change	17:43
	Trigonometric Derivatives	26:58
	The Chain Rule	23:47
	Inverse Trigonometric Functions	27:05
	Equation of a Tangent Line	15:52
Section 4: Derivatives, part 2
	Implicit Differentiation	30:05
	Higher Derivatives	13:16
	Logarithmic and Exponential Function Derivatives	17:42
	Hyperbolic Trigonometric Function Derivatives	14:30
	Related Rates	29:05
	Linear Approximation	23:52
Section 5: Application of Derivatives
	Absolute Minima and Maxima	18:57
	Mean Value Theorem and Rolle's Theorem	20:00
	First Derivative Test, Second Derivative Test	27:11
	L'Hopital's Rule	23:09
	Curve Sketching	40:16
	Applied Optimization	25:37
	Newton's Method	25:13
Section 6: Integrals
	Approximating Areas and Distances	36:50
	Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus	22:02
	Substitution Method for Integration	23:19
Section 7: Application of Integrals, part 1
	Area Between Curves	19:59
	Volume by Method of Disks and Washers	24:22
	Volume by Method of Cylindrical Shells	30:29
	Average Value of a Function	16:31
Section 8: Extra
	Graphs of f, f', f''	23:58
	Slope Fields for Differential Equations	18:32
	Separable Differential Equations	17:04

Related Books

	Calculus by Larson, 9th Edition Authors: Ron Larson, Bruce H. Edwards ISBN: 0547167024 Publisher: Brooks Cole Year: 2009 The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 1 Author: Ron Larson ISBN: 547213093 Publisher: Brooks Cole Year: 2008 This solutions manual includes worked out solutions to every odd-numbered exercise in Calculus of a Single Variable, 9th edition.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 2 Authors: Ron Larson, Bruce H. Edwards ISBN: 547213107 Publisher: Brooks Cole Year: 2009 This solutions manual includes worked out solutions to every odd-numbered exercise in Multivariable Calculus, 9th edition.

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Professor Switkes

The Chain Rule

Slide Duration:

Table of Contents

Section 1: Overview of Functions

Review of Functions

26m 29s

Intro