Partial derivative evaluated at optimum of function

Question

If I take the partial derivative evaluated at the optimum of the function $f(\cdot)$ with respect to $x$, written as $$\frac{\partial f(x,y^*(x))}{\partial x} \quad \textrm{or} \quad \frac{\partial f(x,y)}{\partial x}\biggr\rvert_{y=y^*(x)}$$ where $y^*(x)$ is the value of the $y$ at the optimum, then do I take the partial derivative with respect to $x$, holding $y$ fixed, and then evaluate at $y(x)$, or do I first evaluate at $y(x)$ and then take the partial derivative with respect $x$?

Answer 1

You take the partial derivative with respect to $x$ only. Notice even the name implies it - partial.

BTW - why do you call this $y^*(x)$? The optimal argument $y^*$ is independent of $x$, unless you perform some sort of partial optimization. But in that case you have converted $x$ to a parameter, not a variable, and you find an optimal $y$ for different functions that depend on a pre-defined $x$ value.

If you have a specific function to share it might help to present it.