Derivative Rules: Sum, Difference, Constant Multiple Practice

Sample questions

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1. Sample 1difficulty 1/5

   If $f(x) = 5x^4$, then $f'(x) =$

   • A

     $20x^3$

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   • B

     $20x^4$

   • C

     $x^4/5$

   • D

     $5x^3$

   Why

   $5 \cdot 4 x^3 = 20 x^3$.

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2. Sample 2difficulty 1/5

   If $f(x) = 3x^2 + 5x - 4$, then $f'(x) =$

   • A

     $6x + 5$

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   • B

     $3x + 5$

   • C

     $6x - 4$

   • D

     $3x^2 + 5$

   Why

   Differentiate term-by-term: $6x + 5 + 0 = 6x + 5$.

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3. Sample 3difficulty 1/5

   If $f$ and $g$ are differentiable and $h(x) = 4 f(x) - 3 g(x)$, then $h'(x) =$

   • A

     $f g'$

   • B

     $f - g$

   • C

     $4 f' - 3 g'$

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   • D

     $f' g'$

   Why

   Linearity: derivative distributes through sums and constants.

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4. Sample 4difficulty 1/5

   $\dfrac{d}{dx}(7x) =$

   • A

     $7$

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   • B

     $0$

   • C

     $1$

   • D

     $x$

   Why

   Constant times $x$: derivative is the constant.

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5. Sample 5difficulty 1/5

   If $f(x) = x^3 - 4x + 7$, then $f'(2) =$

   • A

     $12$

   • B

     $8$

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   • C

     $-4$

   • D

     $0$

   Why

   $f'(x) = 3x^2 - 4$. $f'(2) = 12 - 4 = 8$.

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