Calculus Help

Name: The Chain Rule Section:

3.6 The Chain Rule Vocabulary Examples

Chain Rule For two differentiable functions f and g, dd x f (g(x)) =

Alternately, if y is a function of u and u is a function of x,

then dy d x =

1. Each of the following functions is written in the form f (g(x)). For each function, identify f (x) and g(x).

(a) h(x) = √

2x − 1 (b) h(x) = 1

x−1 +1 x−1

(c) h(x) = (

3 5 x − 17

)3/7 (d) h(x) = sin x

Determine the derivative of the following functions.

2. y = (3x − 2)6 3. sin5(x) 4. y = (3x2 + 1)5

5. y = (

x 7 +

7 x

)7 6. y =

(√ x − 3x5

)1/7 7. y = tan(sec x)

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Name: The Chain Rule Section:

8. y = cos(πx + 1) 9. f (x) = −6 sin−3(x) 10. f (x) = 1 sin2 x

11. f (x) = 3 √

3x5 − 1x 12. f (x) = sin(cos x)3

x−7 13. f (x) =

√ cos(3x3 + 2x−1/2)

14. Use trigonometric identities to show that the chain rule applies when determining the derivative of the function f (x) = sin(2x)

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Name: Derivatives of Trigonometric Functions Practice Section:

A.5 Derivatives of Trigonometric Functions Practice

Find dy d x for the following functions.

1. y = x2 − sec x + 1 2. y = 3 csc x + 5x 3. y = x − x 3 sin x

4. y = sec xx 5. y = x 2 cot x 6. y = sin xtan x

7. y = sin2 x 8. y = cos3 x 9. y = −13 (2 + sin 2 x) cos x

Find d2y d x2

for the following functions.

10. y = x sin x − cos x 11. y = 1x + tan x 12. y = sec 2 x

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Name: Derivatives of Trigonometric Functions Practice Section:

13. Determine all of the x-values on the graph of f (x) = −3 sin x cos x for which the tangent line is horizontal.

14. A mass on a spring bounces up and down in simple harmonic motion, modeled by the function d(t) = −15 cos t where d is measured in centimeters and t is measured in seconds. Find the rate at which the spring is oscillating at t = 5 seconds.

Use trigonometric identities to help find the derivative of the following functions.

15. y = sin(2x) 16. y = cos(2x) 17. y = sin ( π 2 − x

)

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Name: Chain Rule Practice Section:

A.6 Chain Rule Practice

Find dy d x for the following functions.

1. y = sin(3x) 2. y = (3x − 6)6 3. y = (3x2 + 3x − 1)4

4. y = 1 (5−2x)2

5. y = cos3(πx) 6. y = sin−2 x

7. y = √

6 + sin(πx2) 8. y = cos( √

4x3 + 5x − 2) 9. y = 1tan(2−x1/5)

10. Let y = ( f (x))3 and suppose that f ′(1) = 4 and

dy d x = 10 for x = 1. Determine f (1).

11. Let y = (

f (x) + 5x2 )4

and suppose that

f (−1) = −4 and dyd x = 3 when x = −1. Determine f ′(−1).

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Name: Chain Rule Practice Section:

12. Determine the equation of the line tangent to y = cos( √

2πx) at x = π2

13. The total cost to produce x boxes of Thin Mint Girls Scout Cookies is C dollars, where C(x) = 0.0001x3 − 0.02x2 + 3x + 300. In t weeks, the production is estimated to be x = 1600 + 100t boxes.

(a) Determine the marginal cost C′(x).

(b) Use Leibniz’ notation for the chain rule, dCdt = dC d x ·

d x dt to determine the rate with respect to time

t that cost is changing. (c) Use your answer to (b) to determine how fast costs are increasing when t = 2 weeks.

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