PhD candidate in Economics · Paris

Derivatives Practice: Upper Intermediate

Harder than the other two sheets: chain rule throughout, and the second half asks for the second derivative. Compute the first derivative for 1 to 10 and the second derivative for 11 to 20, without a calculator.

Work each one out on paper first, then open the solution to check it. The theory is in the derivatives refresher; the other sheets are on the resources page.

  1. 1.
    \( \displaystyle f(x) = (x^2+3)^5 \)
    Solution
    \( \displaystyle f'(x) = 10 x \left(x^{2} + 3\right)^{4} \)
  2. 2.
    \( \displaystyle f(x) = \ln(4x^2+1) \)
    Solution
    \( \displaystyle f'(x) = \frac{8 x}{4 x^{2} + 1} \)
  3. 3.
    \( \displaystyle f(x) = \frac{e^x}{x^2+1} \)
    Solution
    \( \displaystyle f'(x) = \frac{\left(x - 1\right)^{2} e^{x}}{\left(x^{2} + 1\right)^{2}} \)
  4. 4.
    \( \displaystyle f(x) = \sqrt{x} \cdot \ln(x) \)
    Solution
    \( \displaystyle f'(x) = \frac{\ln(x) + 2}{2 \sqrt{x}} \)
  5. 5.
    \( \displaystyle f(x) = x^{1/4} \cdot e^{x^2} \)
    Solution
    \( \displaystyle f'(x) = \frac{\left(8 x^{2} + 1\right) e^{x^{2}}}{4 x^{\frac{3}{4}}} \)
  6. 6.
    \( \displaystyle f(x) = \frac{x^3+2x}{\sqrt{x}} \)
    Solution
    \( \displaystyle f'(x) = \frac{5 x^{2} + 2}{2 \sqrt{x}} \)
  7. 7.
    \( \displaystyle f(x) = \frac{\ln(x)}{x^2} \)
    Solution
    \( \displaystyle f'(x) = \frac{1 - 2 \ln(x)}{x^{3}} \)
  8. 8.
    \( \displaystyle f(x) = \frac{(2x+1)^3}{x^2+1} \)
    Solution
    \( \displaystyle f'(x) = \frac{8 x^{4} + 18 x^{2} + 22 x + 6}{\left(x^{2} + 1\right)^{2}} \)
  9. 9.
    \( \displaystyle f(x) = e^{\ln(x^2+1)} \)
    Solution
    \( \displaystyle f'(x) = 2 x \)
  10. 10.
    \( \displaystyle f(x) = x \cdot \ln(x^2+1) \)
    Solution
    \( \displaystyle f'(x) = \frac{2 x^{2}}{x^{2} + 1} + \ln(x^{2} + 1) \)
  11. 11.
    \( \displaystyle f(x) = x^2 \cdot \ln(x) \)
    Solution
    \( \displaystyle f''(x) = 2 \ln(x) + 3 \)
  12. 12.
    \( \displaystyle f(x) = e^{2x} \cdot x \)
    Solution
    \( \displaystyle f''(x) = 4 x e^{2 x} + 4 e^{2 x} \)
  13. 13.
    \( \displaystyle f(x) = \frac{1}{x} \cdot \ln(x) \)
    Solution
    \( \displaystyle f''(x) = \frac{2 \ln(x) - 3}{x^{3}} \)
  14. 14.
    \( \displaystyle f(x) = (x^2+1)^3 \)
    Solution
    \( \displaystyle f''(x) = 30 x^{4} + 36 x^{2} + 6 \)
  15. 15.
    \( \displaystyle f(x) = \sqrt{x^3+2} \)
    Solution
    \( \displaystyle f''(x) = \frac{3x\left(x^{3} + 8\right)}{4\left(x^{3} + 2\right)^{\frac{3}{2}}} \)
  16. 16.
    \( \displaystyle f(x) = x \cdot e^{-x} \)
    Solution
    \( \displaystyle f''(x) = \frac{x - 2}{e^{x}} \)
  17. 17.
    \( \displaystyle f(x) = \ln(x^2+3x+2) \)
    Solution
    \( \displaystyle f''(x) = - \frac{2 x^{2} + 6 x + 5}{\left(x + 1\right)^{2} \left(x + 2\right)^{2}} \)
  18. 18.
    \( \displaystyle f(x) = \frac{e^x}{x} \)
    Solution
    \( \displaystyle f''(x) = \frac{x^{2} e^{x} - 2 x e^{x} + 2 e^{x}}{x^{3}} \)
  19. 19.
    \( \displaystyle f(x) = x^3 \cdot \ln(x) \)
    Solution
    \( \displaystyle f''(x) = 6 x \ln(x) + 5 x \)
  20. 20.
    \( \displaystyle f(x) = \frac{\ln(x)}{x+1} \)
    Solution
    \( \displaystyle f''(x) = \frac{2 x^{2} \ln(x) - 3 x^{2} - 4 x - 1}{x^{2} \left(x + 1\right)^{3}} \)