# Partial Derivatives of polar co-ordinates

I am getting very confused when trying to find the partial derivative operators in polar co-ordinates. For example, I need to show that $$\partial_{x}=\partial_{r}cos(\theta)-\frac{sin(\theta)}{r}\partial_{\theta}$$, given that $$x=rcos\theta, y=rsin\theta$$

I started by using the chain rule to give that $$\partial_x=\partial_r\frac{\partial r}{\partial x}+\partial_\theta\frac{\partial \theta}{\partial x}$$ and then I went to fine the two partials I could and I am pretty sure this is where I went wrong but I don't understand why it doesn't work. $$x=rcos\theta\\\frac{\partial x}{\partial r}=cos\theta\\\frac{\partial r}{\partial x}=\frac{1}{cos\theta}$$

and similarly for $$\theta$$ I got that $$\frac{\partial \theta}{\partial x}=\frac{-1}{rsin\theta}$$ But I know what I should get and this doesn't get me there. I looked online and saw things with people using the fact that $$tan\theta=\frac{y}{x}$$ but I don't understand why my way doesn't seem to work and I am just getting confused.

As to why $$\frac{\partial r}{\partial x}\ne\frac{1}{\frac{\partial x}{\partial r}}$$ this question may be helpful (Partial derivatives inverse question).
Now to find $$\partial_{x}$$ and $$\partial_{y}$$, you need to use the inverse transformations $$r=\sqrt{x^2+y^2}$$ and $$\theta=arctan(\frac{y}{x})$$
if we differentiate the expression for $$r$$, with respect to $$x$$ and $$y$$ we get $$\frac{\partial r}{\partial x}=\frac{x}{\sqrt{x^2+y^2}}=\frac{rcos(\theta)}{r}=cos\theta$$ and $$\frac{\partial r}{\partial y}=\frac{y}{\sqrt{x^2+y^2}}=\frac{rsin(\theta)}{r}=sin\theta$$
and if we differentiate the expression for $$\theta$$, with respect to $$x$$ and $$y$$ we get $$\frac{\partial \theta}{\partial x}=\frac{1}{1+\frac{y^2}{x^2}}(\frac{-y}{x^2})=-\frac{y}{x^2+y^2}=-\frac{rsin(\theta)}{r^2}=-\frac{sin\theta}{r}$$ $$\frac{\partial \theta}{\partial x}=\frac{1}{1+\frac{y^2}{x^2}}(\frac{1}{x})=\frac{x}{x^2+y^2}=\frac{rcos(\theta)}{r^2}=\frac{cos\theta}{r}$$ you then substitute these expressions in to the chain rule expressions you have. Hopefully that helps