Procedures for Calculating Derivatives Practice

Sample questions

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1. Sample 1difficulty 3/5

   If $f(x) = (x^2 + 1)(x - 3)$, then $f'(x) =$

   • A

     $x^2 + 1 + (x - 3)(2x)$

   • B

     $2x + 1$

   • C

     $2x(x-3)$

   • D

     $3x^2 - 6x + 1$

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   Why

   Use product rule: $2x(x-3) + (x^2+1)(1) = 2x^2 - 6x + x^2 + 1 = 3x^2 - 6x + 1$.

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2. Sample 2difficulty 4/5

   If $y = x^x$, then $\dfrac{dy}{dx}$ at $x = 1$ equals

   • A

     $e$

   • B

     $0$

   • C

     $\ln 2$

   • D

     $1$

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   Why

   $\ln y = x\ln x$. Differentiate to get $y'/y = \ln x + 1$, hence $y' = x^x(\ln x + 1)$. At $x=1$: $y' = 1 \cdot (0+1) = 1$.

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