4.3.2 Using the Chain Rule Flashcards Preview

AP Calculus AB > 4.3.2 Using the Chain Rule > Flashcards

Flashcards in 4.3.2 Using the Chain Rule Deck (14):
1

chain rule

- 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.

2

note

- 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.

3

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

f′(x)=−1/9x^2

4

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

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

5

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

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)

6

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

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

7

Find the derivative.
p(x) = x^4 (2x + 1)^2

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

8

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

f′(x)=(2x)^3⋅6(3x)+(3x)^2⋅6(2x)^2

9

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

f′(x)=24x(2x+3)^3−3(2x+3)^4 / (3x)^2

10

Find the derivative.
p(x) = (3x^)2 (2x + 1)^3

p′(x) = (3x) (2x + 1)^2 (30x + 6)

11

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

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

12

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

y = 1

13

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

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

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)?

y=5/4x+5/16

Decks in AP Calculus AB Class (190):