Section 3.4: The Chain Rule

Find the derivative of the following functions: 1–11.

  1. (a) \( f(x) = (2x - 5)^4 \)   (b) \( f(x) = (1 - x^2)^7 \)   (c) \( f(x) = \sqrt{x^2 - 5x + 1} \)   (d) \( f(x) = \sqrt[3]{x^2 - x + 2} \)  solution

  2. (a) \( f(x) = \dfrac{1}{(3x - 1)^4} \)   (b) \( f(x) = \dfrac{3}{\sqrt{4x^2 + 3}} \)  solution

  3. \( f(x) = (x^2 + 1)^3 (5x + 1)^2 \)  solution

  4. \( f(x) = \dfrac{1 - x^2}{\sqrt{2x + 7}} \)  solution

  5. \( f(x) = \left( x - \dfrac{1}{x} \right)^2 \)  solution

  6. \( f(x) = \left( \dfrac{x^2 - 5}{x^2 + 5} \right)^4 \)  solution

  7. (a) \( f(t) = e^{\tan t} \)   (b) \( f(\theta) = \sin(\cos \theta) \)   (c) \( y = \sin(\tan(5x)) \)  solution

  8. (a) \( y = 5\cot(3\theta) \)   (b) \( f(x) = \sec^2(\tan x) \)   (c) \( f(x) = \tan(e^{2t}) \)  solution

  9. (a) \( f(x) = e^{\frac{\tan x}{x^2}} \)   (b) \( f(\theta) = \sqrt{\cos \theta} \)   (c) \( f(x) = \sin^2(x^2) \)  solution

  10. (a) \( y = \cot^2(\cos\theta) \)   (b) \( y = e^{5t\sin(3t)} \)  solution

  11. (a) \( f(x) = 2^x \)   (b) \( f(t) = 5^{\sin t} \)   (c) \( g(\theta) = 3^{\theta \tan\theta} \)  solution

  12. If \( f(0) = \dfrac{\pi}{4} \), \( f'(0) = \sqrt{2} \), and \( g(x) = \sin(f(x)) \), find \( g'(0) \).  solution

  13. Find an equation of the tangent line to the curve \( y = e^{\cos x} \) at \( x = \dfrac{\pi}{2} \).  solution

  14. Find an equation of the tangent line to the curve \( y = \sin x + \sin^2 x \) at \( (0,0) \).  solution

  15. Find an equation of the tangent line to the curve \( y = \dfrac{6}{2 + e^{-x}} \) at \( (0,2) \).  solution

  16. Given \( F(x) = f(g(x)) \) and \( G(x) = g(f(x)) \), use the table to determine (i) \( F'(1) \) and (ii) \( G'(0) \).  solution

    \( x \) \( f(x) \) \( f'(x) \) \( g(x) \) \( g'(x) \)
    0 0 -2 4 1
    1 0 3 1 2

  17. Find the first and second derivatives of the functions.  solution

    (a) \( f(x) = 3x^4 - 5x^2 + 1 \)   (