Question

Find the derivative of $\cos x$ by first principle.

Answer 1

$f^{{}^{\prime }}\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
$=\underset{h\to 0}{lim}\frac{\cos (x+h)-\cos x}{h}$
$=\underset{h\to 0}{lim}\left[\frac{\cos x\cos h-\sin x\sin h-\cos x}{h}\right]$
$=\underset{h\to 0}{lim}\left[\frac{-\cos x(1-\cos h)-\sin x\sin h}{h}\right]$
$=\underset{h\to 0}{lim}[\frac{-\cos x(1-\cos h)}{h}=\frac{\sin x\sin h}{h}]$
$=-\cos x\left(\underset{h\to 0}{lim}\frac{1-\cos h}{h}\right)-\sin x\underset{h\to 0}{lim}\left(\frac{\sin h}{h}\right)$
$=-\cos x.\left(0\right)-\sin x\left(1\right)=-\sin x$