Say I'm given a function
$\displaystyle u(x/r, 1/r)$
the function u itself is not defined.
when i take a partial derivative with respect to x, do i just write it as
$\displaystyle u_x(x/r, 1/r)$
or is there a chain rule that i have to follow?
Say I'm given a function
$\displaystyle u(x/r, 1/r)$
the function u itself is not defined.
when i take a partial derivative with respect to x, do i just write it as
$\displaystyle u_x(x/r, 1/r)$
or is there a chain rule that i have to follow?
indeed there is a chain rule.
let $\displaystyle s = \frac xr$ and $\displaystyle t = \frac 1r$, then you want $\displaystyle \frac {\partial}{\partial x} u(s,t)$.
by the chain rule, $\displaystyle \frac {\partial u}{\partial x} = \frac {\partial u}{\partial s} \cdot \frac {\partial s}{\partial x} + \frac {\partial u}{\partial t} \cdot \frac {\partial t}{\partial x}$