I have a function $$ q: \mathrm{R}^{N} \to \mathrm{R}^{M} $$ where $M < N$, then another function $$ f:\mathrm{R}^{M} \to \mathrm{R} $$ Let $z = q(\psi)$, I am interested in the derivative $\nabla_{\psi}f(z)$.
My limited understanding of calculus suggests I should think of something like
$$ \nabla_{\psi}f(z) = \nabla_{z}f(z)\nabla_{\psi}q(\psi) $$ But $\nabla_{z}f(z)$ will give an object in $\mathrm{R}^{M}$, and $\nabla_{\psi}q(\psi)$ will give an object in $\mathrm{R}^{N}$, so I don't know how to think about this. I assume I have misunderstood how the derivative should be working in this context. Please could you show me how to take this derivative? n.b. I'm not familiar with the notation used in the Spivak book that has been referenced in answers to other similar questions.
If $$ q:\psi=(\psi_1,\ldots,\psi_N)\in\mathbb{R}^N\to(q_1,\ldots,q_M)\in\mathbb{R}^M $$ and $$ f:z=(z_1,\ldots,z_M)\in\mathbb{R}^M\to\mathbb{R} $$ then $$ \frac{\partial(f\circ q)}{\partial\psi_i}=\sum_{j=1}^N\left(\frac{\partial f}{\partial z_j}\circ q\right)\frac{\partial q_j}{\partial\psi_i},\qquad i=1,\ldots,N, $$ that in matrix form can be shortened as $$ \nabla f(q(\psi)) = \nabla f(z)\nabla q(\psi) $$