Consider the following:
$$ x( r(x,y ) , \theta(x,y) ) = x$$
At 31:18 of this video lecture by Pavel Grinfeld, He says to differentiate both sides with $'x'$ (*) but what kind of derivative with $'x'$ are we doing? The partial or the regular? Also explain why we choose the one we choose (whether it be partial or regular)
(*) : After differentiating with $x$ , the result is
$$ x_{r} r_{x} + x_{\theta} \theta_{x} = 1$$
Note: subscript denote derivatives.
As he is at pains to point out in the video, you need to be careful about the meaning of the symbols. Sometimes $x$ and $y$ are functions, and sometimes co-ordinates, depending on context within the equations. He's talking about the transformation from cartesian co-ordinates $(x,y)$ to polar co-ordinates $(r,\theta)$, and the inverse transformation from polar co-ordinates $(r,\theta)$ back to cartesian co-ordinates $(x,y)$. The forward transformation is a function $P: \mathbb R^2 \to \mathbb R^2$ defined by $P(x,y) = (r(x,y),\theta(x,y))$, where $r$ and $\theta$ are functions defined as $$ r(x,y) = \sqrt{x^2 + y^2}$$ $$\theta(x,y) = arctan(y/x)$$ and the inverse transformation is defined as $P^{-1}(r,\theta) = (x(r,\theta),y(r,\theta))$, where here $x$ and $y$ are functions defined as $$x(r,\theta) = r \cos \theta$$ $$y(r,\theta) = r \sin \theta$$ If we compose the forward and inverse transformations, we get the identity $$P^{-1} \circ P(x,y) = (x,y)$$ or, as functions $$x(r(x,y),\theta(x,y)) = x$$ $$y(r(x,y),\theta(x,y)) = y$$ In the video he goes on to calculate the Jacobeans of $P$, $P^{-1}$ and $P^{-1} \circ P$, which as these are all functions from $\mathbb R^2$ to $\mathbb R^2$ will be 2x2 matrices formed from the partial derivatives of the co-ordinate functions. Now differentiating this function $x$ with respect to the co-ordinate $x$, and using the chain rule, we get $$\frac{\partial x}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial x}{\partial \theta}\frac{\partial \theta}{\partial x}= 1$$ which is easily verified by working out each of the partial derivatives and expressing everything in terms of $r$ and $\theta$.