# Computing partial derivatives for a function from $\mathbb{R}^3$ to $\mathbb{R}^3$

Let $$f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$$ be defined as $$(u, v, w)=f(x,y,z) = (e^{2y} + e^{2z}, e^{2x} - e^{2z}, x - y)$$ I have shown that it has differentiable inverse in a neighborhood of every point in $$\mathbb{R}^3$$. I want to compute $$\frac{\partial x}{\partial u}, \frac{\partial x}{\partial v}, \frac{\partial x}{\partial w}$$ in terms of $$x,y,z$$. Based on the first part, I applied the Inverse Function Theorem and computed the inverse Jacobian $$\partial f^{-1}(u, v, w) = \frac{1}{2(e^{2x} + e^{2y})}\begin{pmatrix} 1 & 1 & 2e^{2y} \\ 1 & 1 & -2e^{2x} \\ e^{2x - 2y} & -e^{2y - 2z} & 2e^{2x+ 2y - 2z} \end{pmatrix}$$ However, I don't know how to proceed. What is $$\frac{\partial x}{\partial u}$$ anyway? Is it going to be number or a vector? All I can compute is $$\frac{\partial f}{\partial x}$$ and $$\frac{\partial f^{-1}}{\partial u}$$. Also the inverse Jacobian isn't even in terms of $$u, v, w$$. As you can tell I am really struggling to understand multivariable differentiation so any help would be much appreciated.

Hint: Use $$u=e^{2y}+e^{2z}$$ $$v=e^{2x}-e^{2z}$$ $$w=x-y$$ throughout, in this manner, for example $$u+v=e^{2x}+e^{2y}$$, etcetera.