Calculating Higher-Order Derivatives Practice

Sample questions

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1. Sample 1difficulty 2/5

   For $f(x) = 2x^4 - x^2$, $f''(x) =$

   • A

     $24x^2 - 2$

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   • B

     $8x^3$

   • C

     $24x^2$

   • D

     $8x^3 - 2x$

   Why

   $f'(x) = 8x^3 - 2x$. $f''(x) = 24 x^2 - 2$.

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2. Sample 2difficulty 2/5

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

   • A

     $96x^2$

   • B

     $0$

   • C

     $24x$

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   • D

     $24$

   Why

   $f'' = 12 x^2$, $f''' = 24 x$.

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3. Sample 3difficulty 2/5

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

   • A

     $24$

   • B

     $24x$

   • C

     $4x^3$

   • D

     $12x^2$

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   Why

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

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4. Sample 4difficulty 3/5

   A linear function has $f''$ equal to:

   • A

     $f$

   • B

     $f'$

   • C

     $0$

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   • D

     Constant nonzero

   Why

   Linear $f$ has constant $f'$, so $f'' = 0$.

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5. Sample 5difficulty 3/5

   For $f(x) = \sin x$, the 100th derivative $f^{(100)}(x)$ equals

   • A

     $\cos x$

   • B

     $-\sin x$

   • C

     $-\cos x$

   • D

     $\sin x$

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   Why

   Derivatives cycle every 4: $\sin, \cos, -\sin, -\cos$. $100 \equiv 0 \pmod 4$, so back to $\sin x$.

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