Inverse function theorem verification

Question

Can it be concluded from $$f^{-1'}(x) = \frac{1}{f'(f^{-1}(x))}$$ that $$f'(x) = \frac{1}{f^{-1'}(f(x))}?$$ The reason I ask is because I have a function that I don't know the derivative of very easily, but I know its inverse and I know the derivative of its inverse, and this is about the only time I've ever used the inverse function theorem so I just want to make sure it works.