4.3.2 Using the Chain Rule

Question 1

chain rule

Answer 1

• The chain rule states that if f ( x ) = g ( h ( x )) , where g and h are differentiable functions, then f is differentiable and f ′ ( x ) = g ′ ( h ( x )) ⋅ h ′ ( x ) .
• Some functions are actually combinations of other functions, such as products or quotients. To differentiate these functions, it may be necessary to use several
  computational techniques and to use some more than once.

Question 2

note

Answer 2

• The chain rule is a shortcut for finding the
  derivative of a composite function.
• To use the chain rule, consider the inside function
  as a single “blop.”
• Take the derivative of the outside function. Then
  replace the “blop” with what it was originally.
  Finally, multiply the entire expression by the
  derivative of the inside (or “blop”).
• Sometimes you will need to use the chain rule in the
  middle of some other rule, such as the product rule
  or quotient rule. Here is one example.
• Work the problem like you normally would, chanting
  through the product rule. If the product rule requires
  you to take the derivative of a composite function,
  then use the chain rule to find that derivative.
• It is also possible to use the chain rule more than
  once on a single problem. Expect to do this when
  taking the derivative of a function that is the
  composition of more than two other functions.

Question 3

Find the derivative of:f(x)=x^3 / (3x^2)^2

Answer 3

f′(x)=−1/9x^2

Question 4

Find the derivative.f(x)=(3x^2+7x)^4 / (2x^3−6x)^3

Answer 4

f′(x)=4(2x^3−6x)^3(3x^2+7x)^3(6x+7) / (2x^3−6x)^6 − 3(3x^2+7x)^4(2x^3−6x)^2(6x^2−6)(2x^3−6x)^6

Question 5

Find the derivative of:

P(t)=(3t^2/3−6t^1/3)^3⋅(3t^2−6t)^1/3

Answer 5

P′(t)=2(3t^2/3−6t^1/3)^3(3t^2−6t)^−2/3(t−1)+ 6(3t^2−6t)^1/3(3t^2/3−6t^1/3)2(t^−1/3−t^−2/3)

Question 6

Find the derivative of:P(t)=(3t^2/3−6t^1/3)^3/(3t^2−6t)^1/3

Answer 6

P′(t)= 6(3t^2−6t)^1/3(3t^2/3−6t^1/3)^2(t^−1/3−t^−2/3) / (3t^2−6t)^2/3 − 2(3t^2/3−6t^1/3)^3(3t^2−6t)^−2/3(t−1) / (3t^2−6t)^2/3

Question 7

Find the derivative.

p(x) = x^4 (2x + 1)^2

Answer 7

p′(x) = 4x^4 (2x + 1) + 4x^3 (2x + 1)^2

Question 8

Find the derivative of f(x) using the product and chain rules:
f(x = (2x)^3 ⋅ (3x)^2

Answer 8

f′(x)=(2x)^3⋅6(3x)+(3x)^2⋅6(2x)^2

Question 9

Find the derivative.f(x)=(2x+3)^4 / 3x

Answer 9

f′(x)=24x(2x+3)^3−3(2x+3)^4 / (3x)^2

Question 10

Find the derivative.

p(x) = (3x^)2 (2x + 1)^3

Answer 10

p′(x) = (3x) (2x + 1)^2 (30x + 6)

Question 11

Suppose f(x)={[x^2+(1/x^2+1)]^3−3x}. Find f′(x).

Answer 11

f′(x)=3[x^2+(1/x^2+1)]^2⋅[2x−2x/(x^2+1)^2]−3

Question 12

Suppose f(x)=(2x^2−4x+3)^4. What is the equation of the line tangent to f at the point (1,1)?

Answer 12

y = 1

Question 13

Suppose f(x)=(x2^+3x−1)^2 / (x^2−3x+4)^2.Find f′(x).

Answer 13

f′(x)=(x^2+3x−1)[(x^2−3x+4)(4x+6)−(x^2+3x−1)(4x−6)] / (x^2−3x+4)^3

Question 14

Suppose f(x)=(x^2+4)^2 / (x^2−2x−3)^2.What is the equation of the line tangent to fat (1,25/16)?

Answer 14

y=5/4x+5/16