Derivatives Using Rules: Power Rule

Formula derivation (video)
  1. Find \(f'(x)\) of each of the following functions. solution
    (a) \( f(x) = x^2 \)   (b) \( f(x) = x^5 \)   (c) \( f(x) = x^{1.8} \)   (d) \( f(x) = x^{\pi} \)
  2. Find \(f'(x)\) of each of the following functions. solution
    (a) \( f(x) = \sqrt{x} \)   (b) \( f(x) = \sqrt[5]{x} \)   (c) \( f(x) = x^{3/4} \)
  3. Find \(f'(x)\) of each of the following functions. solution
    (a) \( f(x) = \dfrac{1}{x} \)   (b) \( f(x) = \dfrac{1}{x^5} \)   (c) \( f(x) = \dfrac{5}{x^{1/3}} \)
  4. Find \(f'(x)\) of each of the following functions. solution
    (a) \( f(x) = 5x^2 + \dfrac{3}{x^2} \)   (b) \( f(x) = 4\sqrt{x} - 10x + 7 \)   (c) \( f(x) = \dfrac{1}{x^{4/3}} - \dfrac{x^3}{5} + 7\pi \)
  5. Find the derivative of the following functions. solution
    (a) \( f(r) = e^r + r^e \)   (b) \( f(x) = x(x^2 - 5) + e^5 \)
  6. Find the derivative of the following functions. solution
    (a) \( f(s) = \sqrt{s}(s - 1) \)   (b) \( f(x) = \dfrac{\sqrt{x} + x}{x^2} \)
  7. If \( f(t) = \dfrac{t^2 - 3t + 1}{\sqrt{t}} \), find \( f'(t) \). solution
  8. (a) If \( f(x) = x^3 - 4x^2 + 5\pi \), find \( f'(-2) \).
    (b) If \( f(x) = x^4 + 2e^x \), find \( f'(0) \). solution
  9. (a) If \( f(x) = \sqrt{x} \), find \( f'(4) \).
    (b) If \( g(x) = \dfrac{1}{\sqrt[3]{x}} \), find \( g'(1) \). solution
  10. Find an equation of the tangent line to the curve \( y = x^4 - 3x^2 + 5 \) at \( x = 1 \). solution
  11. Find an equation of the tangent line to the curve \( y = \sqrt[4]{x} - x \) at the point \( (1, 0) \). solution
  12. The equation of motion of a particle is \( s = t^4 - 2t^3 + t^2 - t \).
    (a) Find the velocity after 1 second.
    (b) Find the acceleration after 1 second. solution
  13. Find the points on the curve \( y = 2x^3 + 3x^2 - 12x + 1 \) where the tangent is horizontal. solution
  14. For what value of \( x \) does the graph of \( f(x) = e^x - 2x \) have a horizontal tangent line? solution
  15. Find the first and second derivatives of the function. solution
    (a) \( f(x) = x^4 - 3x^3 + 7x \)   (b) \( f(x) = \dfrac{1}{x^5} \)   (c) \( f(x) = \sqrt[5]{x} \)
  16. Find the third derivative of the given function. solution
    (a) \( f(x) = 2x^5 + 8x^2 - 3x + 5 \)   (b) \( f(x) = \dfrac{1}{x} \)