Does the chain rule work for trigonometric functions?

df/dt = df/dp * dp/dq * dq/dt

If f(t) = Cos^2(wt)   <--- supposed to be cosine-squared (cosine(wt) * cosine(wt))

f(p) = p^2

p(q) = Cos(q)

q(t) = wt


Is this correct?

d/dt [Cos^2(wt)] = 2*Cos(wt) * (-Sin(wt)) * (w)


Now that I have typed this out I am certain it is true and that the chain rule works just fine for trig functions.   Please confirm for me, or tell me I stupid  but then give the correct answer.



ETA - I just noticed the effective frequency of Cos^2(wt) is double that of Cos(wt).

Looks correct, but it's been 12 years since my last calc class.

Thank you, guys.