Answer
$f'(x)=-\sin x$
Work Step by Step
Need to prove $f'(x)=-\sin x$
$f'(x)=\lim\limits_{h \to 0}\dfrac{f(x+h)-f(x)}{h}$
since, we have $f(x)=\cos x$, then
$f'(x)=\lim\limits_{h \to 0}\dfrac{\cos (x+h)-\cos (x)}{h}\\=\lim\limits_{h \to 0}\dfrac{\cos x \cos h-\sin x \sin h-\cos x}{h}\\=\lim\limits_{h \to 0}\dfrac{\cos x \cos h-\cos x-\sin x \sin h}{h}\\=\cos x [\lim\limits_{h \to 0}\dfrac{ \cos h-1}{h}]-\sin x [\lim\limits_{h \to 0}\dfrac{ \sin h}{h}]\\=\cos x (0)-\sin x(1)\\\\=-\sin x$
