4.3.1 An Introduction to the Chain Rule

Question 1

intro to chain rule

Answer 1

• A composite function is made up of layers of functions inside of functions. Some techniques of differentiation become very cumbersome when applied to composite
  functions.
• 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 ) .

Question 2

notes

Answer 2

• A composite function is a function that results
  from applying a function to the results of another
  function.
• Each different function that is applied can be
  thought of as a layer of the composite function.
• To find the derivative of a composite function, you
  must look at each layer.
• The chain rule is a shortcut for finding the
  derivative of a composite function. The chain rule
  must be used for each layer of the composite
  function.
• The chain rule states that the derivative of a
  composition of two functions is equal to the
  derivative of the outer function evaluated at the
  inner function times the derivative of the inner
  function.
• Consider the inside of the composite function as a
  “blop.” Take the derivative of that piece as though
  the “blop” was just x. Then multiply that result by
  the derivative of the “blop.”
• Notice that the chain rule can simplify the process of
  finding some derivatives.

Question 3

Suppose f(x)=(x^2−6)^4 / 8. Find f′(2).

Answer 3

−16

Question 4

Find the derivative of f(x).f(x)=(x^3+5x+1)^3

Answer 4

f′(x)=3(x3+5x+1)^2(3x^2+5)

Question 5

Find the derivative of:f(x)=3√x^4/3+x^1/3

Answer 5

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

Question 6

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

Answer 6

f′(x)=−6x[x^2−(1+x^2)^2]^2⋅[1+2x^2]

Question 7

f(x)=3(4x+7)^4−4(3x+7)^3

Answer 7

f′(x)=48(4x+7)^3−36(3x+7)^2

Question 8

Find the derivative of:

h(x) = (3x + 5)^4

Answer 8

h’(x)=12(3x+5)^3

Question 9

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

Answer 9

f′(1)=0

Question 10

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

Answer 10

f′(x)=2[2x−√1+x^2]⋅[2−x√1+x^2]

Question 11

Find the derivative of f(x).

f(x)=7(x^7/3+11/7x^7/5+13x^7/7)4/3

Answer 11

f′(x)=28/3(x^7/3+11/7x^7/5+13x)^1/3⋅(7/3x^4/3+11/5x^2/5+13)

Question 12

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

Answer 12

f′(x)=6x^5+36x^3+48x

Question 13

Find the derivative.f(x)=(x^2+1)^2

Answer 13

f′(x)=2(x^2+1)(2x)

Question 14

Find the derivative of P(t).P(t)=−(2t^2−14t+4)^2

Answer 14

P′(t)=−2(2t^2−14t+4)(4t−14)

Question 15

Find the derivative.f(x)=(3x)^11

Answer 15

f′(x)=33(3x^)10