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. 

Well if you find trouble remembering them maybe you can use the formulas $$\cos{x} = \frac{e^{ix} + e^{-ix}}{2} \quad \sin{x} = \frac{e^{ix} - e^{-ix}}{2i}$$ 
which you can get from Euler's formula and then if you forget what the derivative of $\cos{x}$ or of $\sin{x}$ is then you can just differentiate those exponentials, which is rather easy. 
And I guess that you will not have problems remembering what the derivative of the exponential is. 
Also using the power series representations for the sine and the cosine you can differentiate them term by term and verify easily that $(\cos{x})' = -\sin{x}$ and $(\sin{x})' = \cos{x}$.
But in any case, depending on how you define the trigonometric functions, there may be different ways to prove that each derivative is what it is. 
I hope this helps a little.
