How to find the derivative of cos^2x? I seem to be confused.

I was told to use the chain rule, but I don't see how it can apply.

Can't I just substitute (cos x)(cos x) in its place and then use the product rule? Isn't cos^2 x = (cos x)(cos x)?

Please let me know if I am right. If not, then please explain how to use the chain rule in...; How are you getting: ?