AP®MathsCollege BoardCalculus ABExam QuestionsUnit 3: Differentiation of Composite, Implicit & Inverse FunctionsDifferentiation of Composite & Inverse Functions

Differentiation of Composite & Inverse Functions (College Board AP® Calculus AB): Exam Questions

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Differentiation of Composite & Inverse Functions

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1
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If y equals 5 space sin open parentheses x over 5 close parentheses, then fraction numerator d squared y over denominator d x squared end fraction equals

  • negative 125 sin open parentheses x over 5 close parentheses

  • 1 fifth sin open parentheses x over 5 close parentheses

  • negative sin open parentheses x over 5 close parentheses

  • negative 1 fifth sin open parentheses x over 5 close parentheses

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2
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If f open parentheses x close parentheses equals e to the power of square root of x end exponent, then f apostrophe open parentheses x close parentheses equals

  • square root of x e to the power of open parentheses square root of x minus 1 close parentheses end exponent

  • fraction numerator e to the power of square root of x end exponent over denominator 2 square root of x end fraction

  • fraction numerator e to the power of square root of x end exponent over denominator square root of x end fraction

  • e to the power of square root of x end exponent

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3
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What is the slope of the line tangent to the curve y equals arctan space open parentheses 2 x close parentheses at the point at which x equals 1 half?

  • 1

  • 1 half

  • pi over 4

  • 2 over 3

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4
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Let f be a differentiable function with an inverse function f to the power of negative 1 end exponent.

If f open parentheses 3 close parentheses equals 5 and f apostrophe open parentheses 3 close parentheses equals 4 what is open parentheses f to the power of negative 1 end exponent close parentheses to the power of apostrophe open parentheses 5 close parentheses?

  • 1 third

  • 1 fourth

  • 4

  • 3

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5
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If y equals open parentheses 4 x cubed plus 5 x plus 2 close parentheses to the power of 5, then fraction numerator d y over denominator d x end fraction equals

  • 5 open parentheses 4 x cubed plus 5 x plus 2 close parentheses to the power of 4

  • open parentheses 12 x squared plus 5 close parentheses to the power of 5

  • 5 open parentheses 12 x squared plus 5 close parentheses open parentheses 4 x cubed plus 5 x plus 2 close parentheses to the power of 4

  • open parentheses 12 x squared plus 5 close parentheses open parentheses 4 x cubed plus 5 x plus 2 close parentheses to the power of 4

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1
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If y equals open parentheses x to the power of 4 plus sin x close parentheses to the power of 5, then y to the power of apostrophe equals

  • 5 open parentheses x to the power of 4 plus sin x close parentheses to the power of 4

  • 5 open parentheses 4 x cubed plus cos x close parentheses to the power of 4

  • 5 open parentheses 4 x cubed plus cos x close parentheses to the power of 4 times open parentheses 12 x squared minus sin x close parentheses

  • 5 open parentheses x to the power of 4 plus sin x close parentheses to the power of 4 times open parentheses 4 x cubed plus cos x close parentheses

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2
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Let f open parentheses x close parentheses equals open parentheses 3 x minus 1 close parentheses cubed and let g be the inverse function of f. Given that f open parentheses 0 close parentheses equals negative 1, what is the value of g to the power of apostrophe open parentheses negative 1 close parentheses?

  • negative 1 fourth

  • 1 over 16

  • 1 over 9

  • 9

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3
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If f open parentheses x close parentheses equals cos space open parentheses e to the power of negative 2 x end exponent close parentheses, then f to the power of apostrophe open parentheses x close parentheses equals

  • negative sin open parentheses e to the power of negative 2 x end exponent close parentheses

  • negative sin open parentheses e to the power of negative 2 x end exponent close parentheses minus 2 e to the power of negative 2 x end exponent

  • negative 2 e to the power of negative 2 x end exponent space sin open parentheses e to the power of negative 2 x end exponent close parentheses

  • 2 e to the power of negative 2 x end exponent space sin open parentheses e to the power of negative 2 x end exponent close parentheses

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4
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If f open parentheses x close parentheses equals tan open parentheses 2 x close parentheses, then f to the power of apostrophe open parentheses pi over 2 close parentheses equals

  • 2

  • 0

  • negative 2

  • 1

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5
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A function f is defined by f open parentheses x close parentheses equals arccos open parentheses 2 x close parentheses.

What is the second derivative of f open parentheses x close parentheses?

  • negative fraction numerator 8 x over denominator open parentheses square root of 1 minus 4 x squared end root close parentheses cubed end fraction

  • negative fraction numerator 2 over denominator square root of 1 minus 4 x squared end root end fraction

  • fraction numerator 8 x over denominator open parentheses square root of 1 minus 4 x squared end root close parentheses cubed end fraction

  • fraction numerator 2 over denominator square root of 1 minus 4 x squared end root end fraction

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6
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Let f be the function defined by f open parentheses x close parentheses equals 4 x squared plus 2 x for x greater than negative 1 fourth. If g left parenthesis x right parenthesis equals f to the power of negative 1 end exponent open parentheses x close parentheses and g open parentheses 20 close parentheses equals 2, what is the value of g apostrophe open parentheses 20 close parentheses?

  • 1 over 162

  • 1 over 18

  • 18

  • 162

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1
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If f open parentheses x close parentheses equals square root of x squared minus 13 end root and g open parentheses x close parentheses equals 4 x plus 1, then the derivative of f open parentheses g open parentheses x close parentheses close parentheses at x equals 1 is

  • fraction numerator 10 over denominator square root of 3 end fraction

  • fraction numerator 20 over denominator square root of 3 end fraction

  • fraction numerator 36 over denominator square root of 3 end fraction

  • fraction numerator 20 over denominator square root of 5 end fraction

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2
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fraction numerator d over denominator d x end fraction open parentheses e to the power of e to the power of x end exponent close parentheses equals

  • e to the power of e to the power of x end exponent

  • x e to the power of e to the power of x end exponent

  • e to the power of e to the power of x plus x end exponent

  • e to the power of e to the power of x squared end exponent end exponent

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3
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If f open parentheses x close parentheses equals x cubed plus 3 x, then fraction numerator d over denominator d x end fraction open parentheses f open parentheses ln open parentheses x squared close parentheses close parentheses close parentheses equals

  • fraction numerator 6 open parentheses open parentheses ln space x close parentheses squared plus 1 close parentheses over denominator x end fraction

  • fraction numerator 3 open parentheses ln space x close parentheses squared plus 3 over denominator x end fraction

  • 3 open parentheses ln space x close parentheses squared plus 3

  • fraction numerator 6 open parentheses 4 open parentheses ln space x close parentheses squared plus 1 close parentheses over denominator x end fraction

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4
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Let f be a differentiable function such that f open parentheses 2 close parentheses equals negative 1, f open parentheses 3 close parentheses equals 2, f apostrophe open parentheses 2 close parentheses equals negative 3 and f apostrophe open parentheses 3 close parentheses equals negative 2.

The function g is differentiable and g open parentheses x close parentheses equals f to the power of negative 1 end exponent open parentheses x close parentheses for all x.

What is the value of g apostrophe open parentheses 2 close parentheses ?

  • negative 1 third

  • negative 1

  • negative 1 half

  • 3

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5
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If f open parentheses x close parentheses equals ln open parentheses 2 x minus 3 plus e to the power of negative x end exponent close parentheses, then f to the power of apostrophe apostrophe end exponent open parentheses 0 close parentheses equals

  • negative 1 third

  • negative 3 over 4

  • negative 1 half

  • negative 1 fourth

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6
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An equation of the line tangent to the graph of f open parentheses x close parentheses equals x open parentheses 2 minus 3 x close parentheses to the power of 4 at the point open parentheses 1 comma space 1 close parentheses is

  • y equals negative 12 x plus 13

  • y equals negative 11 x plus 12

  • y equals 11 x minus 10

  • y equals 13 x minus 12

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7
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If f open parentheses x close parentheses equals open parentheses x plus 3 close parentheses open parentheses x squared minus 1 close parentheses cubed, then f apostrophe left parenthesis x right parenthesis equals

  • 6 x open parentheses x squared minus 1 close parentheses squared

  • 6 x open parentheses x plus 3 close parentheses open parentheses x squared minus 1 close parentheses squared

  • open parentheses x squared minus 1 close parentheses squared open parentheses x squared plus 3 x plus 8 close parentheses

  • open parentheses x squared minus 1 close parentheses squared open parentheses 7 x squared plus 18 x minus 1 close parentheses

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