Partial derivative of a function of a function Suppose $z=f(x+y)$. Taking partial derivatives w.r.t. $x$ on either side:
$\partial z/\partial x = \partial f(x+y)/\partial x$. Let us denote $x+y$ by $u$. So $\partial z/\partial x = \partial f(u(x,y))/\partial x$. How do I evaluate the R.H.S. The chain rule I know is for finding total derivatives and not partial derivatives.
 A: The correct formula is
$$\frac{\partial f\bigl(u(x,y)\bigr)}{\partial x}=\frac{\mathrm d\bigl(f\bigl(u(x,y)\bigr)\bigr)}{\mathrm d u}\cdot\frac{\partial \bigl(u(x,y)\bigr)}{\partial x}=\frac{\mathrm d\bigl(f\bigl(u(x,y)\bigr)\bigr)}{\mathrm d u}\cdot 1$$
A: If $z$ (or $f$) depends on a single variable $u$, which in turn depends on two variables $x$ and $y$, then $z$ is a function of those two "ultimately" independent variables $x$ and $y$, and by (a multivariate version of) the Chain Rule:
$$\frac{\partial z}{\partial x}=\frac{\mathrm{d}z}{\mathrm{d}u}\cdot\frac{\partial u}{\partial x} \quad \text{and} \quad \frac{\partial z}{\partial y}=\frac{\mathrm{d}z}{\mathrm{d}u}\cdot\frac{\partial u}{\partial y}.$$
Although this is not in your question, but let me point out that there are many versions of multivariate functions, because we can combine functions of different numbers of variables in all kinds of different ways. For example, if $z$ depends on two variables $u$ and $v$, both of which in turn depend on a single variable $x$, then $z$ is a function of that one "ultimately" independent variable $x$ only, and the appropriate version of the Chain Rule is:
$$\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{\partial z}{\partial u}\cdot\frac{\mathrm{d}u}{\mathrm{d}x}+\frac{\partial z}{\partial v}\cdot\frac{\mathrm{d}v}{\mathrm{d}x}.$$
The graphs below can you help understand multivariate chain rules for these two examples, as well as assist in setting it up correctly for any other composition of functions:

Arrows indicate how functions depend on variables. And to set up the correct chain rule, you find all trajectories from the top-level function to the same bottom-level variable and add together "chain rules" along those paths.
