Let $M$ be a differentiable sub-manifold of $\mathbb{R}^n$. Let $N$ be a a differentiable sub-manifold of $\mathbb{R}^s$ and $\psi:N\to M$ a map of $C^k$ class. Prove $\nabla(f\circ\Psi)(y)=(\Psi'(y))^{-1}(\nabla(f)(\Psi(y)))\:\:\forall y\in N$.
I know $\nabla(h\circ f)(y)=h'(y)\nabla(f,x)\:\:\forall x\in M$. However I cannot see how to prove the statement. How come $\nabla(f\circ\Psi)(y)=(\Psi'(y))^{-1}(\nabla(f)(\Psi(y)))$?
Question:
Can someone provide me a proof?