I'm having trouble grasping the concept which proves that the derivative of $f(x) = \cos(x)$ is $f'(x) = -\sin(x)$. It needs to be proven using the definition of a derivative--and I can't quite piece it together in my head. Could somebody clarify the concept for me? I appreciate it very much. I'd just like to re-iterate that it needs to be proven using the definition of a derivative. Thanks again.
I'd appreciate as many steps as possible--also, if you could define some of the laws you are using, that'd be very helpful. My trigonometry skills are very mediocre. I appreciate the help you all are giving, but I'm still very cloudy on my understanding.