Question

Find the derivative by first principle
$\frac{1}{2x+3}$

Answer 1

Let $f\left(x\right)=\frac{1}{2x+3}$

$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lim}\frac{\frac{1}{2(x+h)+3}-\frac{1}{2x+3}}{h}=\underset{h\to 0}{lim}\frac{\frac{-2h}{(2x+2h+3)(2x+3)}}{h}$
$={lim}_{h\to 0}\frac{-2}{(2x+2h+3)(2x+3)}$

$=\frac{-2}{(2x+3{)}^2}$