Is there any intuition behind why the derivative of $\cos(x)$ is equal to $-\sin(x)$ or just something to memorize? why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i'm missing, or is it just one of those things to memorize? and i know this is not very differential equation related, just one of those things that I've never really understood. 
and alliteratively why is $$ \int\sin(x)dx=-\cos(x)$$
any good explanations would be greatly appreciated. 

I am with Henning Makholm on this and produce a sketch

The slope of the blue line is the red line, and the slope of the red line is the green line.  I know the blue line is sine and the red line is cosine; the green line can be seen to be the negative of sine.  Similarly the partial area under the red line is the blue line, though this sort of approach leads to $\int \sin(x)dx=1-\cos(x)$.
