# Multivariable partial derivative

1. Dec 7, 2015

### RichardJ

1. The problem statement, all variables and given/known data
From the transformation from polar to Cartesian coordinates, show that

\frac{\partial}{\partial x} = \cosφ \frac{\partial}{\partial r} - \frac{\sinφ}{r} \frac{\partial}{\partialφ}

2. Relevant equations
The transformation from polar to Cartesian coordinates is assumed to be x = r\cosφ

3. The attempt at a solution
To solve the problem i tried to use the multivariable chain rule. Resulting in the following equation:

\frac{\partial}{\partial x} =\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partialφ}{\partial x}\frac{\partial}{\partial φ}

Writing $r = x/\cosφ$ and $\arccos(x/r) = φ$ i tried to solve this problem. But this does not give the right answer.

Am i using the right approach? I think it is necessary to use the multivariable chain rule in some form. But the partial derivative not acting on some other function seems a bit weird to me so i am not sure how to solve this problem.

Last edited: Dec 7, 2015
2. Dec 7, 2015

### Ray Vickson

In LaTeX, standard functions look a lot better if they are preceded by '\', so you get $\sin \phi$ instead of $sin \phi$, etc.

3. Dec 7, 2015

### Samy_A

With that equation in mind:
$r=\sqrt{x²+y²}$
$φ=\arctan(\frac{y}{x})$ (with some subtleties).

4. Dec 8, 2015

### RichardJ

Ahh, thanks a lot. That solved the problem.