Formular for partial gradient of multi-variable function?

Question

Is there a property of gradient that allow me to compute $\partial f/\partial x$ with $f=f(x, y, z)$ and $y$ and $z$ are functions of $x$.Can we write that $\partial f/\partial x = \partial f/\partial y * \partial y / \partial x + \partial f/\partial z *\partial z / \partial x$ or something like that. I came upon this question when trying to understand the backpropagation equation of recurrent neural network and I don't understand while it sum all the partial gradient through time.

Answer 1

If $f$ is a function of $(x,y,z)$ and $(y,z)$ are themselves functions of $x$, then $f$ can be rewritten so that it is a function of $x$ alone. Let's call the single-variable function $\phi(x)$.

So there are two distinct quantities which one might call the derivative (gradient) of the function wrt $x$ $$\eqalign{ &\frac{\partial f}{\partial x} \cr {\rm -or-} \cr &\frac{d\phi}{dx} &= \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx} + \frac{\partial f}{\partial z} \frac{dz}{dx} }$$ where $\partial$ has been used for the partial derivative of multi-variable functions, and $d$ for the derivative of a single-variable function.