Why is the derivative of cosx = -sinx?

Had an opportunity to look at this today with a student. Notice that if f(x) = cosx, f'(x) = -sinx. Why?

Well, remember that the derivative is the slope. So when we take the derivative at x=0 of cosx, it is obviously 0 (since the tangent line is horizontal). Now, notice that as you take the slope of cosx from x=0 to x=pi, the slope is negative. So f'(x) starts at 0 and gets smaller until you reach x=pi. After that, the slope becomes positive. Thus, you are tracing out a sin curve, but it is flipped (-sinx). Easy thing to figure out if you happen to forget that the derivative of cosx is -sinx.