The Product Rule Practice

Sample questions

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1. Sample 1difficulty 2/5

   $\dfrac{d}{dx}\bigl[u(x) v(x)\bigr] =$

   • A

     $u v$

   • B

     $u' + v'$

   • C

     $u' v + u v'$

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   • D

     $u' v'$

   Why

   Product rule (Leibniz form).

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2. Sample 2difficulty 2/5

   If $f(x) = x^2 \sin x$, then $f'(x) =$

   • A

     $x^2 \cos x$

   • B

     $2x \sin x$

   • C

     $x^2 \cos x + 2x \sin x$

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   • D

     $2x \cos x$

   Why

   $(x^2)' \sin x + x^2 (\sin x)' = 2x \sin x + x^2 \cos x$.

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3. Sample 3difficulty 3/5

   By the product rule, $\dfrac{d}{dx}[fg]$ equals:

   • A

     $\dfrac{f'g - fg'}{g^2}$

   • B

     $f'g'$

   • C

     $f'g - fg'$

   • D

     $f'g + fg'$

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   Why

   Product rule: $(fg)' = f'g + fg'$.

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4. Sample 4difficulty 3/5

   If $f(x) = e^x \sin x$, then $f'(x) =$

   • A

     $e^x \cos x$

   • B

     $e^x \sin x \cos x$

   • C

     $e^x(\cos x - \sin x)$

   • D

     $e^x(\sin x + \cos x)$

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   Why

   $e^x \sin x + e^x \cos x = e^x(\sin x + \cos x)$.

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