Section 3.4: The Chain Rule

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  16. Refer the table for the questions (a) and (b).

    17. \(x\)\( f(x) \)\( f'(x) \)\( g(x) \)\( g'(x) \)
    0  0 -241
    1 0 312

    (a)  If \(  F(x)= f(f(x)), \) find \( F'(1)=\)   solution     (b)  If \(  G(x)= g(f(x)), \) find \( G'(0)=\)  solution 

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

    18. (a) \(\displaystyle{f(x)= 3x^4-5x^2+1} \)    (b) \(\displaystyle{f(x)= (x+1)^4} \)     (c) \(\displaystyle{f(x)= (1- x^2)^7}\)