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)

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Differentiation of Composite & Inverse Functions

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Differentiation of Composite & Inverse Functions

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 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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3
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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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4
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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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5
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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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6
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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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