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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

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1
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If f open parentheses x close parentheses equals cos space x, then f apostrophe open parentheses pi over 3 close parentheses equals

  • negative fraction numerator square root of 3 over denominator 2 end fraction

  • negative 1 half

  • 1 half

  • fraction numerator square root of 3 over denominator 2 end fraction

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2
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If y equals e to the power of x over x, then fraction numerator d y over denominator d x end fraction equals

  • e

  • fraction numerator e to the power of x open parentheses x minus 1 close parentheses over denominator x squared end fraction

  • fraction numerator e to the power of x open parentheses x plus 1 close parentheses over denominator x squared end fraction

  • 0

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3
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If f open parentheses x close parentheses equals x open parentheses 3 x minus 7 close parentheses, then f apostrophe open parentheses x close parentheses equals

  • negative 7

  • 3

  • 3 x minus 7

  • 6 x minus 7

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4
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If y equals sec space x, then fraction numerator d y over denominator d x end fraction equals

  • negative csc squared space x

  • negative cot space x space csc space x

  • tan space x space sec space x

  • sec squared space x

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5
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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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1
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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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2
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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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3
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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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4
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Let f and g be differentiable functions with the following properties:

f open parentheses x close parentheses greater than 0 for all x

g open parentheses 5 close parentheses equals 3

If h open parentheses x close parentheses equals f open parentheses x close parentheses g open parentheses x close parentheses and h apostrophe open parentheses x close parentheses equals f apostrophe open parentheses x close parentheses g open parentheses x close parentheses, then g open parentheses x close parentheses equals

  • g apostrophe open parentheses x close parentheses

  • f open parentheses x close parentheses

  • 3

  • 5

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5
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If f open parentheses x close parentheses equals tan open parentheses 3 x close parentheses, then f apostrophe open parentheses pi over 18 close parentheses equals

  • square root of 3

  • 2 square root of 3

  • 4

  • 12

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1
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What is the instantaneous rate of change at x equals 3 of the function f given by f open parentheses x close parentheses equals fraction numerator x squared minus 5 over denominator x minus 4 end fraction?

  • negative 10

  • negative 2

  • negative 1 over 10

  • 10

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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 f left parenthesis x right parenthesis equals x square root of 2 x end root, which of the following is not correct.

  • f apostrophe open parentheses 1 half close parentheses equals 3 over 2

  • f apostrophe open parentheses 1 close parentheses equals fraction numerator 3 square root of 2 over denominator 2 end fraction

  • f apostrophe open parentheses 2 close parentheses equals fraction numerator 2 square root of 2 over denominator 3 end fraction

  • f apostrophe open parentheses 3 close parentheses equals fraction numerator 3 square root of 6 over denominator 2 end fraction

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4
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Let f be a differentiable function with f open parentheses 3 close parentheses equals 18 and f apostrophe open parentheses 3 close parentheses equals 9, and let g open parentheses x close parentheses equals ln space x space f open parentheses x close parentheses. Which of the following is an equation of the line tangent to the graph of g at the point where x equals 3?

  • y minus 18 equals 9 open parentheses x minus 3 close parentheses

  • y minus 19.8 equals 15.9 open parentheses x minus 3 close parentheses

  • y minus 19.8 equals 9 open parentheses x minus 3 close parentheses

  • y minus 19.8 equals negative 3.89 open parentheses x minus 3 close parentheses

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5
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For which value of x do the graphs of y equals open parentheses ln space x close parentheses squared and y equals square root of x plus 1 over e to the power of x have parallel tangent lines?

  • 0.726

  • 1.079

  • 1.544

  • 2.346

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