The Quotient Rule Practice

Sample questions

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1. Sample 1difficulty 2/5

   $\dfrac{d}{dx}\dfrac{u}{v} =$

   • A

     $\dfrac{u'}{v} - \dfrac{u v'}{v^2}$

   • B

     $\dfrac{u' v - u v'}{v^2}$

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   • C

     $\dfrac{u'}{v'}$

   • D

     $\dfrac{u' v + u v'}{v^2}$

   Why

   Quotient rule.

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2. Sample 2difficulty 3/5

   Use the quotient rule on $\tan x = \sin x/\cos x$: $\dfrac{d}{dx}\tan x =$

   • A

     Both B and C

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   • B

     $\sec x$

   • C

     $1 + \tan^2 x$

   • D

     $\sec^2 x$

   Why

   $(\cos x \cdot \cos x - \sin x \cdot (-\sin x))/\cos^2 x = 1/\cos^2 x = \sec^2 x = 1 + \tan^2 x$.

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3. Sample 3difficulty 3/5

   If $f(x) = \dfrac{x}{x+1}$, then $f'(x) =$

   • A

     $\dfrac{-1}{(x+1)^2}$

   • B

     $\dfrac{x}{(x+1)^2}$

   • C

     $\dfrac{1}{(x+1)^2}$

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   • D

     $1$

   Why

   Quotient rule: $\dfrac{1 \cdot (x+1) - x \cdot 1}{(x+1)^2} = \dfrac{1}{(x+1)^2}$.

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4. Sample 4difficulty 3/5

   By the quotient rule, $\dfrac{d}{dx}\left[\dfrac{f}{g}\right]$ equals:

   • A

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

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   • B

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

   • C

     $\dfrac{f'}{g'}$

   • D

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

   Why

   Quotient rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.

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5. Sample 5difficulty 4/5

   $\dfrac{d}{dx}\dfrac{\sin x}{x}$ at $x = \pi/2$ is

   • A

     $0$

   • B

     $-\frac{4}{\pi^2}$

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   • C

     $1$

   • D

     $\frac{2}{\pi}$

   Why

   Quotient: $(x\cos x - \sin x)/x^2$. At $x=\pi/2$: $(\pi/2 \cdot 0 - 1)/(\pi^2/4) = -1/(\pi^2/4) = -4/\pi^2$.

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