Find $\dfrac{d}{dx}(\cos x)$
I know the answer is $-\sin x$ only by process of elimination. I can find solution graphically but I need to know algebraically. Here is my proof so far.
$\begin{align*} \dfrac{d}{dx}\cos x=\lim_{h\to 0}\dfrac{\cos (x+h)-\cos x}{h} &=\lim_{h\to 0}\dfrac{\cos x\cos h-\sin x\sin h-\cos x}{h} \end{align*}$
And that's where I end up and I have no clue where to go from here. Can someone please give me the next step but not the complete answer.