21 Homework: Power Rule

  1. Compute the following derivatives. Do not use the definition of the derivative. Instead, use the linearity and power rules we talked about in this section.
    1. [latex]\cfrac{d}{dx}\ x^{15}[/latex]
      [latex]15 x^{14}[/latex]

      ans
    2. [latex]\cfrac{d}{dx}\ 3x^6[/latex]
      [latex]18x^5[/latex]

      ans
    3. [latex]\cfrac{d}{dx}\ \cfrac{1}{2}x^{4}[/latex]
      [latex]2x^3[/latex]

      ans
    4. [latex]\cfrac{d}{dx}\ 3 x^2 + 6x - 1[/latex]
      [latex]6x + 6[/latex]

      ans
    5. [latex]\cfrac{d}{dx}\ (2x + 3)^2[/latex]
      [latex]8x + 12[/latex]

      ans
    6. [latex]\cfrac{d}{dx}\ x^{1/3}[/latex]
      [latex]\frac{1}{3} x^{-2/3}[/latex]

      ans
    7. [latex]\cfrac{d}{dx}\ 7 x^{-4}[/latex]
      [latex]-28x^{-5}[/latex]

      ans
    8. [latex]\cfrac{d}{dx}\ 2x^{-1/2} + 4x^{1/2}[/latex]
      [latex]-x^{-3/2} + 2x^{-1/2}[/latex]

      ans
    9. [latex]\cfrac{d}{dx}\ \sqrt{x}[/latex]
      [latex]\frac{1}{2}x^{-1/2}[/latex]

      ans
    10. [latex]\cfrac{d}{dx}\ \frac{1}{x}[/latex]
      [latex]-x^{-2}[/latex]

      ans
    11. [latex]\cfrac{d}{dx}\ \sqrt[3]{x^2}[/latex]
      [latex]\frac{2}{3} x^{-1/3}[/latex]

      ans
    12. [latex]\cfrac{d}{dx}\ \frac{2}{\sqrt{x}}[/latex]
      [latex]-x^{-3/2}[/latex]

      ans

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