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AP Calculus AB: 4.3.3 Combining Computational Techniques

Mathematics14 CardsCreated 8 months ago

This content explains the chain rule for differentiating composite functions and discusses how it integrates with other differentiation techniques like the product and quotient rules. It highlights the usefulness of different notations, especially Leibniz notation, for clarity and ease when applying the chain rule in complex problems involving multiple function combinations.

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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)=g’(h(x))*h’(x).
...

note

  • There are several ways to describe functions. Often
    the y-notation is used instead of function notation.

  • In function nota...

Find dy/dx: y=(2x+1)^2(3x)^2.

dy/dx=(2x+1)^2(18x)+(3x)^2(8x+4)

Find dy/dx, where y=(x+5)^3(x−3)^3

d/ydx=−24(x+5)^2 / (x−3)^4

A particle’s position is given by the function x(t)=(4−t)^3.What is the value of dx/dt when t=3?

dx/dt=−3

Given y=4x^3 / 3, find dy/dx.

dydx=4x^2

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AP Calculus AB: 4.3.3 Combining Computational Techniques
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

  • There are several ways to describe functions. Often
    the y-notation is used instead of function notation.

  • In function notation, the prime symbol ( ) is often used to indicate a derivative. The prime symbol can also be used when working in y-notation. Notice, however, that
    Leibniz notation has certain advantages over prime notation.

  • The connection between the derivative and slope is more apparent with Leibniz notation.

  • Leibniz notation is also easier to use when remembering some formulas, such as the chain rule.

  • Notice that this function is actually a combination of several other functions. Specifically, the function involves a quotient, a product, and two composite functions.

  • Start with the outermost combination. You can tell which combination to start with by thinking about what you would do last by order of operations.

  • Once you start using the quotient rule, you will need to take the derivative of the numerator. To do so you will need the product rule.

  • The numerator involves a composite function. To find its derivative you will need to use the chain rule.

  • Finding the derivative of the denominator will also require the chain rule.

Find dy/dx: y=(2x+1)^2(3x)^2.

dy/dx=(2x+1)^2(18x)+(3x)^2(8x+4)

Find dy/dx, where y=(x+5)^3(x−3)^3

d/ydx=−24(x+5)^2 / (x−3)^4

A particle’s position is given by the function x(t)=(4−t)^3.What is the value of dx/dt when t=3?

dx/dt=−3

Given y=4x^3 / 3, find dy/dx.

dydx=4x^2

Find dy/dx given that y=3u^2,u=2v,and v=x/6.

dy/dx=2x/3

Given y=(3x^2+7x)^3, find dy/dx.

dy/dx=3(3x^2+7x)^2(6x+7)

Given P=−3t^2+6t, find dP/dt.

dP/dt=−6t+6

If h=3k^2 and k=(3t)^2, then find dh/dt

dh/dt=972t^3

Given y=(x^2+3x)^3(2x^2−4x)^2,find dy/dx.

dy/dx=2(x2+3x)^3(2x^2−4x) (4x−4) + 3(2x^2−4x)^2(x^2+3x)^2(2x+3).

Given y=2x^2, find dy/dx

dy/dx=4x

Find dy/dx given that y=3u^2/3+2u^−1/2, and u=x^2.

dy/dx=4x^1/3−2x^−2

Given y=(2x+1)^2/ 3x^2, find dy/dx.

dy/dx=3x^2⋅4(2x+1)−(2x+1)^2⋅6x / (3x^2)^2