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AP Calculus AB: 4.3.1 An Introduction to the Chain Rule

Mathematics15 CardsCreated 3 months ago

This content introduces the chain rule, a key technique in calculus for differentiating composite functions—functions built from layers of other functions. It explains how to identify inner and outer layers, apply the rule step-by-step, and includes examples demonstrating how the chain rule simplifies otherwise complex differentiation problems.

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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 ) .

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Key Terms

Term
Definition

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 ...

notes

  • A composite function is a function that results
    from applying a function to the results of another
    function.

  • Each diff...

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

−16

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

f′(x)=3(x3+5x+1)^2(3x^2+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]

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]

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AP Calculus AB: 4.3.1 An Introduction to the Chain Rule
TermDefinition

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 ) .

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.

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

−16

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

f′(x)=3(x3+5x+1)^2(3x^2+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]

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]

f(x)=3(4x+7)^4−4(3x+7)^3

f′(x)=48(4x+7)^3−36(3x+7)^2

Find the derivative of:

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

h’(x)=12(3x+5)^3

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

f′(1)=0

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]

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)

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

f′(x)=6x^5+36x^3+48x

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

f′(x)=2(x^2+1)(2x)

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

P′(t)=−2(2t^2−14t+4)(4t−14)

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

f′(x)=33(3x^)10