Applying the Power Rule Practice

Sample questions

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1. Sample 1difficulty 1/5

   If $f(x) = x^7$, then $f'(x) =$

   • A

     $7x^7$

   • B

     $7x^6$

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   • C

     $x^6$

   • D

     $\tfrac{x^8}{8}$

   Why

   Power rule: $\frac{d}{dx}x^n = n x^{n-1}$.

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2. Sample 2difficulty 1/5

   For $f(x) = 6 - 2x + x^2$, $f'(1) =$

   • A

     $1$

   • B

     $2$

   • C

     $0$

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   • D

     $5$

   Why

   $f'(x) = -2 + 2x$. $f'(1) = 0$.

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3. Sample 3difficulty 2/5

   If $f(x) = \sqrt{x}$, then $f'(x) =$

   • A

     $\sqrt x/2$

   • B

     $2\sqrt x$

   • C

     $\tfrac{1}{\sqrt x}$

   • D

     $\tfrac{1}{2\sqrt x}$

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   Why

   $\sqrt x = x^{1/2}$. $f'(x) = \tfrac{1}{2} x^{-1/2} = 1/(2\sqrt x)$.

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4. Sample 4difficulty 2/5

   If $f(x) = x^{-1/3}$, then $f'(x) =$

   • A

     $-\tfrac{1}{3}x^{-4/3}$

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   • B

     $-3 x^{-4/3}$

   • C

     $\tfrac{1}{3}x^{-2/3}$

   • D

     $3x^{-2/3}$

   Why

   Power rule: $-1/3 \cdot x^{-1/3 - 1} = -\tfrac{1}{3} x^{-4/3}$.

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5. Sample 5difficulty 2/5

   $\dfrac{d}{dx}\bigl(x^{2/3}\bigr) =$

   • A

     $\tfrac{2}{3} x^{1/3}$

   • B

     $\tfrac{3}{2} x^{1/3}$

   • C

     $\tfrac{3}{2} x^{-1/3}$

   • D

     $\tfrac{2}{3} x^{-1/3}$

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   Why

   Power rule: $\tfrac{2}{3} x^{2/3 - 1} = \tfrac{2}{3} x^{-1/3}$.

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