AP®MathsCollege BoardCalculus ABExam QuestionsUnit 2: Differentiation Definition & Fundamental PropertiesFundamental Properties of Differentiation

Fundamental Properties of Differentiation (College Board AP® Calculus AB)

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Fundamental Properties of Differentiation

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Fundamental Properties of Differentiation

1
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x

f open parentheses x close parentheses

f to the power of apostrophe open parentheses x close parentheses

g open parentheses x close parentheses

g to the power of apostrophe open parentheses x close parentheses

2

4

-5

-2

3

3

1

2

-3

6

The table above gives values of the differentiable functions f and g and their derivatives at selected values of x. If h is the function defined by h open parentheses x close parentheses equals f open parentheses x close parentheses g open parentheses x close parentheses plus 3 f open parentheses x close parentheses then h to the power of apostrophe open parentheses 2 close parentheses equals

  • 7

  • 17

  • -30

  • 24

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2
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limit as h rightwards arrow 0 of fraction numerator tan space 5 open parentheses x plus h close parentheses space minus space tan open parentheses 5 x close parentheses over denominator h end fraction equals

  • 0

  • 5 sec squared open parentheses 5 x close parentheses

  • sec squared open parentheses 5 x close parentheses

  • 5 cot open parentheses 5 x close parentheses

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3
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If y equals x cos x, then fraction numerator d y over denominator d x end fraction equals

  • cos x minus sin x

  • cos x plus x sin x

  • cos x minus x sin x

  • x open parentheses cos x minus sin x close parentheses

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4
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If f open parentheses x close parentheses equals 6 minus 5 x minus ln x, then f to the power of apostrophe open parentheses 1 close parentheses equals

  • negative 3

  • negative 4

  • negative 5

  • negative 6

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5
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The function f is defined by f open parentheses x close parentheses equals fraction numerator x over denominator x plus 3 end fraction. What points open parentheses x comma space y close parentheses on the graph of f have the property that the line tangent to f at open parentheses x comma space y close parentheses has slope 1 third?

  • open parentheses 0 comma space 0 close parentheses only

  • open parentheses 1 third comma fraction numerator space 1 over denominator 10 end fraction close parentheses only

  • open parentheses 0 comma space 0 close parentheses and open parentheses 6 comma space 2 over 3 close parentheses

  • open parentheses 0 comma space 0 close parentheses and open parentheses negative 6 comma space 2 close parentheses

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