Property 7
Derivative of Division of Two Functions
\(\left[\samefrac{f(x)}{g(x)}\right]'=\samefrac{f'(x)\cdot g(x)-g'(x)\cdot f(x)}{g^2(x)}\)
Note
\((\tan x)'=\left(\samefrac{\sin x}{\cos x}\right)'=\samefrac{\cos x\cdot\cos x+\sin x\cdot\sin x}{\cos^2x}\)
\(=\boxed{\samefrac1{\cos^2x}=\sec^2x=1+\tan^2x}\)
\((\cot x)'=\left(\samefrac{\cos x}{\sin x}\right)'=\samefrac{-\sin x\cdot\sin x-\cos x\cdot\cos x}{\sin^2x}\)
\(=\boxed{-\samefrac1{\sin^2x}=-\operatorname{cosec}^2x=-(1+\cot^2x)}\)
1.
\(f(x)=\samefrac{3x-1}{x+2}\)
\(\Rightarrow f'(x)=?\)
2.
\(f(x)=\samefrac{x+2}{x^2-1}\)
\(\Rightarrow f'(x)=?\)
3.
\(f(x)=\samefrac{x^3+2}{e^x}\)
\(\Rightarrow f'(0)=?\)
4.
\(f(x)=\tan x\)
\(\Rightarrow f'(0)=?\)
5.
\(f(x)=\samefrac{\cos x}{3^x}\)
\(\Rightarrow f'(0)=?\)
6.
\(f(x)=\samefrac{x^2+3x}{2x-1}\)
\(\Rightarrow f'(1)=?\)
7.
\(f(x)=\samefrac{4x-7}{3x+2}\)
\(\Rightarrow f'(1)=?\)
8.
\(f(x)=\samefrac{h(x)}{g(x)}\)
\(h(2)=2\)
\(h'(2)=3\)
\(g(2)=1\)
\(g'(2)=4\)
\(\Rightarrow f'(2)=?\)