PhD candidate in Economics · Paris

Derivatives Practice: Starter

Twenty-four exercises straight from the definition or from the basic rules. Constants, polynomials, two quotients and a run of square roots. Start here if the rules are still new.

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.
    \( \displaystyle g(x) = 10 \)
    Solution
    \( \displaystyle g'(x) = 0 \)
  2. 2.
    \( \displaystyle T(y) = -8 \)
    Solution
    \( \displaystyle T'(y) = 0 \)
  3. 3.
    \( \displaystyle f(x) = 5x + 7 \)
    Solution
    \( \displaystyle f'(x) = 5 \)
  4. 4.
    \( \displaystyle Q(t) = 1 - 12t \)
    Solution
    \( \displaystyle Q'(t) = -12 \)
  5. 5.
    \( \displaystyle f(z) = z^2 + 3 \)
    Solution
    \( \displaystyle f'(z) = 2 z \)
  6. 6.
    \( \displaystyle R(w) = w^2 - 8w + 20 \)
    Solution
    \( \displaystyle R'(w) = 2 w - 8 \)
  7. 7.
    \( \displaystyle V(t) = 6t - t^2 \)
    Solution
    \( \displaystyle V'(t) = 6 - 2 t \)
  8. 8.
    \( \displaystyle Q(t) = 2t^2 - 8t + 10 \)
    Solution
    \( \displaystyle Q'(t) = 4 t - 8 \)
  9. 9.
    \( \displaystyle g(z) = 1 + 10z - 7z^2 \)
    Solution
    \( \displaystyle g'(z) = 10 - 14 z \)
  10. 10.
    \( \displaystyle f(x) = 5x - x^3 \)
    Solution
    \( \displaystyle f'(x) = 5 - 3 x^{2} \)
  11. 11.
    \( \displaystyle Y(t) = 2t^3 + 9t + 5 \)
    Solution
    \( \displaystyle Y'(t) = 6 t^{2} + 9 \)
  12. 12.
    \( \displaystyle Z(x) = 2x^3 - x^2 - x \)
    Solution
    \( \displaystyle Z'(x) = 6 x^{2} - 2 x - 1 \)
  13. 13.
    \( \displaystyle f(t) = 2t - 3 \)
    Solution
    \( \displaystyle f'(t) = 2 \)
  14. 14.
    \( \displaystyle g(x) = \frac{x+2}{1-x} \)
    Solution
    \( \displaystyle g'(x) = \frac{3}{\left(1 - x\right)^{2}} \)
  15. 15.
    \( \displaystyle Q(t) = \frac{t^2}{t+2} \)
    Solution
    \( \displaystyle Q'(t) = \frac{t^{2} + 4 t}{\left(t + 2\right)^{2}} \)
  16. 16.
    \( \displaystyle f(w) = \sqrt{w+8} \)
    Solution
    \( \displaystyle f'(w) = \frac{1}{2 \sqrt{w + 8}} \)
  17. 17.
    \( \displaystyle V(t) = \sqrt{14+3t} \)
    Solution
    \( \displaystyle V'(t) = \frac{3}{2 \sqrt{3 t + 14}} \)
  18. 18.
    \( \displaystyle G(x) = \sqrt{2-5x} \)
    Solution
    \( \displaystyle G'(x) = - \frac{5}{2 \sqrt{2 - 5 x}} \)
  19. 19.
    \( \displaystyle Q(t) = \sqrt{1+4t} \)
    Solution
    \( \displaystyle Q'(t) = \frac{2}{\sqrt{4 t + 1}} \)
  20. 20.
    \( \displaystyle f(x) = \sqrt{x^2+1} \)
    Solution
    \( \displaystyle f'(x) = \frac{x}{\sqrt{x^{2} + 1}} \)
  21. 21.
    \( \displaystyle W(t) = \frac{1}{\sqrt{t}} \)
    Solution
    \( \displaystyle W'(t) = - \frac{1}{2 t^{\frac{3}{2}}} \)
  22. 22.
    \( \displaystyle g(x) = 4\sqrt{1-x} \)
    Solution
    \( \displaystyle g'(x) = - \frac{2}{\sqrt{1 - x}} \)
  23. 23.
    \( \displaystyle f(x) = x + \sqrt{x} \)
    Solution
    \( \displaystyle f'(x) = 1 + \frac{1}{2 \sqrt{x}} \)
  24. 24.
    \( \displaystyle f(x) = x + \frac{1}{x} \)
    Solution
    \( \displaystyle f'(x) = 1 - \frac{1}{x^{2}} \)