Question about Spivaks Inverse function theorem proof In the opening lines of Spivaks "Calculus on Manifolds" proof of the inverse function theorem he writes the equation below. let $\lambda = Df(a)$
$$D(\lambda^{-1}\circ f)(a) = D(\lambda^{-1})(f(a))\circ Df(a) = \lambda^{-1}\circ Df(a)$$
I don't see how $ D(\lambda^{-1})(f(a)) = \lambda^{-1}$ in the second equality
and  in the statment of the theorem makes the standard assumptions of f being $c^1$ and $f'(a)$ being nonsingular.

It's simply because $\lambda$ is a linear map. So, its inverse is also a linear map and the derivative of a linear map at every point of its domain is again that linear map.
