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AP Calculus AB: 4.3.2 Using the Chain Rule

Mathematics14 CardsCreated 3 months ago

This content elaborates on the chain rule, a fundamental technique for differentiating composite functions. It explains how to treat the inner function as a “blop,” differentiate the outer function, then multiply by the derivative of the inner function. It also highlights that the chain rule can be combined with other rules (product, quotient) and used multiple times when dealing with nested compositions.

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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.

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Key Terms

Term
Definition

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 ) =...

note

  • The chain rule is a shortcut for finding the
    derivative of a composite function.

  • To use the chain rule, consider the insi...

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

f′(x)=−1/9x^2

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

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)

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...

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AP Calculus AB: 4.3.2 Using the Chain Rule
TermDefinition

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.

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.

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

f′(x)=−1/9x^2

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

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)

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

Find the derivative.

p(x) = x^4 (2x + 1)^2

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

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

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

f′(x)=24x(2x+3)^3−3(2x+3)^4 / (3x)^2


Find the derivative.

p(x) = (3x^)2 (2x + 1)^3

p′(x) = (3x) (2x + 1)^2 (30x + 6)

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

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

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

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