# Differential Equation, Change of variables

1. Jun 16, 2017

### binbagsss

1. The problem statement, all variables and given/known data
Hi,

I am looking at this question:

With this (part of ) solution:

2. Relevant equations

3. The attempt at a solution

I follow up to the last line-

I do not understand here how we have simply taken the $1/t^{\alpha m + \alpha}$ outside of the derivative $\frac{\partial}{\partial y}$ since $y=y(r,t^{\beta})$ i.e. $t$ and $y$ are not independent variables.... $\frac{\partial}{\partial t}= \frac{\partial y}{\partial t^{\beta}}\frac{\partial t^{\beta}}{\partial t}$

A partial derivative expresses the consequences of changing one variable while one or more other variables are held constant. The full notation specifies what is held constant. E.g. if f=f(x,y) then we may write $\frac {\partial f}{\partial x}\Big\rvert_{y=y_0}$. Nearly always, it is obvious what the 'other' variables are, and we don't need to specify their values, so we omit the vertical bar and its subscript.