4.3.1 An Introduction to the Chain Rule Flashcards Preview

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Flashcards in 4.3.1 An Introduction to the Chain Rule Deck (15):
1

intro to chain rule

- 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 ) .

2

notes

- 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.

3

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

−16

4

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

f′(x)=3(x3+5x+1)^2(3x^2+5)

5

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

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

6

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

f′(x)=−6x[x^2−(1+x^2)^2]^2⋅[1+2x^2]

7

f(x)=3(4x+7)^4−4(3x+7)^3

f′(x)=48(4x+7)^3−36(3x+7)^2

8

Find the derivative of:
h(x) = (3x + 5)^4

h'(x)=12(3x+5)^3

9

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

f′(1)=0

10

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

f′(x)=2[2x−√1+x^2]⋅[2−x√1+x^2]

11

Find the derivative of f(x).
f(x)=7(x^7/3+11/7x^7/5+13x^7/7)4/3

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

12

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

f′(x)=6x^5+36x^3+48x

13

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

f′(x)=2(x^2+1)(2x)

14

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

P′(t)=−2(2t^2−14t+4)(4t−14)

15

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

f′(x)=33(3x^)10

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