Section 3.4: The Chain Rule
Find the derivative of the following functions: 1--11.
(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
(a) \(\displaystyle{f(x)= \frac{1}{(3x-1)^4}} \) (b) \(\displaystyle{f(x)=\frac{3}{\sqrt{4x^2+3}}} \) solution
\(\displaystyle{f(x)= (x^2+1)^3(5x+1)^2} \) solution
\(\displaystyle{f(x)=\frac{1-x^2}{\sqrt{2x+7}}} \) solution
\(\displaystyle{f(x)=\left( x - \frac{1}{x}\right)^2} \) solution
\(\displaystyle{f(x)=\left( \frac{x^2-5}{x^2+5}\right)^4} \) solution
(a) \(\displaystyle{f(t) = e^{\tan t}} \) (b) \(\displaystyle{f(\theta)= \sin(\cos \theta)} \) (c) \(\displaystyle{y= \sin(\tan (5x))} \) solution
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(a) \( y= 5 \cot(3 \theta) \) (b) \( f(x) = \sec^2{(\tan x)} \) (c) \(\displaystyle{f(x)=\tan(e^{2t}) }\) solution
(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
(a) \( \displaystyle{y= \cot^2(\cos(\theta))} \) (b) \(\displaystyle{y=e^{5t\sin(3t)}}\) solution
(a) \(\displaystyle{f(x) = 2^{x}} \) (b) \(\displaystyle{f(t) = 5^{\sin t}} \) (c) \(\displaystyle{g(\theta) = 3^{\theta\tan \theta}} \) solution
If \( f(0)=\dfrac{\pi}{4}, f'(0)=\sqrt{2}\) and \(g(x)=\sin(f(x)) \), find \( g'(0)\). solution
Find an equation of the tangent line to the curve \(y= e^{\cos x} \) at \( x=\frac{\pi}{2}\). solution
Find an equation of the tangent line to the curve \(\displaystyle{ y=\sin x + \sin^2 x}\) at \( (0,0)\). solution
Find an equation of the tangent line to the curve \(\displaystyle{ y=\frac{6}{2+e^{-x}} }\) at \( (0,2)\). solution
Given that \(F(x)=f(g(x))\) 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 |
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) \(\displaystyle{f(x)= (1-x^2)^7}\)