AP®MathsCollege BoardCalculus BCExam QuestionsUnit 3: Differentiation of Composite, Implicit & Inverse FunctionsImplicit Differentiation

Implicit Differentiation (College Board AP® Calculus BC): Exam Questions

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Implicit Differentiation

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1
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1 mark

If x equals sin space y then fraction numerator d y over denominator d x end fraction equals

  • fraction numerator 1 over denominator square root of 1 plus x squared end root end fraction

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

  • fraction numerator 1 over denominator cos space x end fraction

  • fraction numerator 1 over denominator 1 minus x squared end fraction

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2
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1 mark

What points open parentheses x comma space y close parentheses on the graph of x squared plus y squared equals 50 have the property that the line tangent to the graph has slope negative 1?

  • open parentheses 5 comma space minus 5 close parentheses and open parentheses negative 5 comma space 5 close parentheses

  • open parentheses 10 comma space 10 close parentheses and open parentheses negative 10 comma space minus 10 close parentheses

  • open parentheses 10 comma space minus 10 close parentheses and open parentheses negative 10 comma space 10 close parentheses

  • open parentheses 5 comma space 5 close parentheses and open parentheses negative 5 comma space minus 5 close parentheses

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3
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1 mark

If x squared plus 3 y cubed equals 5 open parentheses 2 x plus y close parentheses, then fraction numerator d y over denominator d x end fraction equals

  • table row blank blank cell fraction numerator 10 minus 2 x over denominator 6 y minus 5 end fraction end cell end table

  • table row cell 2 x plus 9 y squared end cell equals 15 end table

  • table row blank blank cell fraction numerator 10 minus 2 x over denominator 9 y squared minus 5 end fraction end cell end table

  • table row blank blank cell fraction numerator 10 plus 5 y over denominator 2 x plus 9 y squared end fraction end cell end table

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4
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1 mark

If e to the power of x plus y end exponent equals 3 x then fraction numerator d y over denominator d x end fraction equals

  • 3 over e to the power of x plus y end exponent

  • e to the power of x plus y end exponent

  • 3 over e to the power of x plus y end exponent plus 1

  • 3 over e to the power of x plus y end exponent minus 1

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5
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1 mark

If ln open parentheses y minus x close parentheses equals x squared then fraction numerator d y over denominator d x end fraction equals

  • 1 minus 2 x squared plus 2 x y

  • 2 x squared plus 2 x y minus 1

  • 2 x y minus 2 x squared plus 1

  • fraction numerator 1 over denominator y minus x end fraction

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1
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1 mark

If negative 3 x squared plus 2 x y equals 12, then when x equals 1, fraction numerator d y over denominator d x end fraction equals

  • negative 9 over 2

  • 9 over 2

  • 15 over 2

  • negative 15 over 2

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2
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1 mark

If the equation of a curve is 2 x minus 4 x y plus 2 y squared equals 4, then at the point (2, 4), fraction numerator d y over denominator d x end fraction is

  • negative 7 over 4

  • 7 over 4

  • 1 fourth

  • negative 11 over 8

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3
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1 mark

Which of the following is not a valid derivative of fraction numerator x squared over denominator 4 y cubed end fraction equals 5?

  • fraction numerator x over denominator 30 y squared end fraction

  • fraction numerator 2 y over denominator 3 x end fraction

  • fraction numerator 8 x y cubed over denominator 12 x squared y squared end fraction

  • fraction numerator 30 y squared over denominator x end fraction

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4
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1 mark

Which of the following is the equation of the tangent to the curve x squared plus 2 y squared minus 4 x equals 8 at the point open parentheses 4 comma space 2 close parentheses?

  • y equals negative 1 half x plus 4

  • y equals negative 1 half x minus 4

  • y equals 1 half x plus 4

  • y equals 1 half x minus 4

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5
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1 mark

If cos open parentheses 2 x minus y close parentheses equals 3 x, fraction numerator d y over denominator d x end fraction equals

  • fraction numerator negative 3 over denominator sin open parentheses 2 x minus y close parentheses end fraction minus 2

  • 2 minus fraction numerator 3 over denominator sin open parentheses 2 x minus y close parentheses end fraction

  • fraction numerator 3 plus 2 sin open parentheses 2 x minus y close parentheses over denominator sin open parentheses 2 x minus y close parentheses end fraction

  • fraction numerator 3 over denominator sin open parentheses 2 x minus y close parentheses end fraction

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1
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1 mark

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

  • 2 over x open parentheses ln space x close parentheses open parentheses x to the power of ln space x end exponent close parentheses

  • 2 x to the power of ln space x end exponent

  • 4 over x open parentheses ln space x close parentheses open parentheses x to the power of ln space x end exponent close parentheses

  • 4 x to the power of ln space x end exponent open parentheses ln space x close parentheses

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2
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1 mark

If x to the power of 4 minus 6 x y squared plus 9 y equals 10, then fraction numerator d y over denominator d x end fraction equals

  • fraction numerator 4 x cubed minus 6 y squared over denominator 12 x y plus 9 end fraction

  • fraction numerator 4 x cubed minus 6 y squared over denominator 12 x y minus 9 end fraction

  • fraction numerator 4 x cubed minus 6 y squared plus 9 over denominator 12 x y end fraction

  • fraction numerator 4 x cubed minus 12 x y minus 6 y squared over denominator 9 end fraction

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3
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1 mark

If y equals tan open parentheses x y close parentheses, then fraction numerator d y over denominator d x end fraction equals

  • fraction numerator y sec squared open parentheses x y close parentheses over denominator 1 minus x sec squared open parentheses x y close parentheses end fraction

  • fraction numerator sec squared open parentheses x y close parentheses over denominator x y end fraction

  • y times sec squared open parentheses x y close parentheses

  • fraction numerator y sec squared open parentheses x y close parentheses over denominator 1 plus x sec squared open parentheses x y close parentheses end fraction

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4
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1 mark

What can be said about the slope of the line tangent to the curve x squared y plus 2 y squared equals 2 plus x squared at the point open parentheses 4 comma space 1 close parentheses

  • Positive slope

  • Negative slope

  • Horizontal

  • Vertical

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5
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1 mark

If y equals arctan open parentheses x plus y close parentheses, fraction numerator d y over denominator d x end fraction equals

  • 1 minus fraction numerator 1 over denominator 1 plus open parentheses x plus y close parentheses squared end fraction

  • fraction numerator open parentheses x plus y close parentheses squared over denominator 1 plus open parentheses x plus y close parentheses squared end fraction

  • fraction numerator 1 over denominator 1 plus open parentheses x plus y close parentheses squared end fraction

  • 1 over open parentheses x plus y close parentheses squared

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