$f:\mathbb{R^2} \rightarrow \mathbb{R}$ is differentiable and I want to find the partial derivatives of $g(x,y)=f(y,x)$
$\frac{\partial g}{\partial x}(x,y) = \lim_{t\rightarrow0} \frac{g((x,y)+t(1,0))-g(x,y)}{t} = \lim_{t\rightarrow0} \frac{g(x+t,y)-g(x,y)}{t}= \lim_{t\rightarrow0} \frac{f(y,x+t)-f(y,x)}{t} = \frac{\partial f}{\partial y}(y,x)$
I understand this, but isn’t the point of finding the partial derivative of $g$ finding it in terms of $f(x,y)$?