# Formular for partial gradient of multi-variable function?

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.

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.