Question 486705
Use the definition of derivative using limits:


*[tex \LARGE \frac{d}{dx} (\cos(x)) = \lim_{\Delta x \to 0} \frac{\cos(x+\Delta x) - \cos(x)}{\Delta x}]


*[tex \LARGE = \lim_{\Delta x \to 0} \frac{\cos x \cos(\Delta x) - \sin x \sin (\Delta x) - \cos x}{\Delta x} = \lim_{\Delta x \to 0} \frac{\cos x (\cos (\Delta x) - 1)}{\Delta x} - \frac{\sin x \sin(\Delta x)}{\Delta x}]


Applying our known facts, we can treat cos x and sin x as constants and say that the first limit collapses to zero, and the second limit collapses to -sin x. Hence,


*[tex \LARGE \frac{d}{dx} (\cos x) = -\sin x]


We could also use L'Hopital's rule to evaluate the limits, but that assumes we already know the derivative of cos x, so we shouldn't use it right now.