Section 3.4: The Chain Rule

A. 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).

\(x\)\( f(x) \)\( f'(x) \)\( g(x) \)\( g'(x) \)
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

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