Question

Find the derivative by first principle
$\cos 5x$

Answer 1

Let $f\left(x\right)=\cos 5x$

$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lim}\frac{\cos 5(x+h)-\cos 5x}{h}=\underset{h\to 0}{lim}\frac{-2\sin (\frac{10x+5h}{2})\sin (\frac{5h}{2})}{h}$

$=(-2\sin 5x){lim}_{h\to 0}\frac{\sin \frac{5h}{2}}{h}=-5\sin 5x$