What is the derivative of the cos(x)

-sin(x)

From
lim(h-->0) (cos h cos x - sin h sin x - cos x) / h,

rewrite it as
lim(h-->0) ((cos h - 1) cos x - sin h sin x) / h
= lim(h-->0) cos x * (cos h - 1)/h - sin x * (sin h / h)
= cos x * 0 - sin x * 1
= -sin x.

In the next to last step, I used the same limits that one would use to derive the derivative formula for y = sin x.

Extra note:
All you really need to assume is lim(h-->0) (sin h)/h = 1.

Then, we see that lim(h-->0) (cos h - 1)/h = 0:
lim(h-->0) (cos h - 1)/h
= lim(h-->0) (cos^2(h) - 1) / [h (cos h + 1)]
= lim(h-->0) -sin^2(h) / [h (cos h + 1)]
= lim(h-->0) (sin h / h) * (-sin h / (cos h + 1))
= 1 * 0/2
= 0.

-sin(x)

- sin(x)