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

Implicit Differentiation (College Board AP® Calculus AB)

Exam Questions

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Implicit Differentiation

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Implicit Differentiation

1
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A cylinder with a diameter of 8 feet contains a liquid. The liquid escapes through a hole in the bottom of the container. The rate of change of the height h of the liquid in the container with respect to time is modelled by fraction numerator d h over denominator d t end fraction equals negative 1 over 25 square root of h, where h is measured in feet and t is measured in minutes.

(The volume of a cylinder with radius r and height h is V equals pi r squared h.)

Find the rate of change of the volume of liquid in the container with respect to time when the height of the liquid is 9 feet. Indicate units of measure.

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2a
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Consider the curve given by the equation 3 y squared minus 12 equals y space cos space x for y greater than 0.

Show that fraction numerator d y over denominator d x end fraction equals fraction numerator y space sin space x over denominator cos space x space minus space 6 y end fraction.

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2b
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Write an equation for the line tangent to the curve at the point open parentheses fraction numerator 3 pi over denominator 2 end fraction comma space 2 close parentheses.

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3a
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Consider the curve given by the equation 8 x y equals 3 plus y to the power of 4.

Show that fraction numerator d y over denominator d x end fraction equals fraction numerator 2 y over denominator y cubed minus 2 x end fraction.

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3b
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A particle is moving along the curve. At the instant when the particle is at the point open parentheses 7 over 2 comma space 3 close parentheses, its horizontal position is increasing at a rate of fraction numerator d x over denominator d t end fraction equals 10 units per second.

What is the value of fraction numerator d y over denominator d t end fraction, the rate of change of the particle's vertical position, at that instant?

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4
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Show that the derivative of y equals arcsin open parentheses 3 x close parentheses is fraction numerator d y over denominator d x end fraction equals fraction numerator 3 over denominator square root of 1 minus 9 x squared end root end fraction.

Show the work that leads to your answer.

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5
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Use implicit differentiation to show that the derivative of y equals x to the power of x is fraction numerator d y over denominator d x end fraction equals x to the power of x open parentheses ln space x space plus thin space 1 close parentheses.

Show all steps of your working.

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