Prove that if the functions $f_1,...f_n$ in the statement of the implicit function theorem are assumed to be $k$ times differentiable (i.e., all partial derivatives of order k exist and are continuous), then the same is true of all the component functions $\varphi_1,...,\varphi_n$ of $\varphi.$
I know I should differentiate $f(x,\varphi(x))=0$ and express the Jacobi matrix of $\varphi$ in terms of the Jacobi matrix of $f$ and then explain why the Jacobi matrix of $\varphi$ is continuously differentiable.