Derivative of a function that's evaluated at a point

Question

If $f(x,y)$ is differentiable then what would $$\frac{\partial}{\partial x} \left[f(x,y)|_{x=a}\right]$$ evaluate to? Would it be $\frac{\partial f}{\partial x}(x,y)|_{x=a}$? Or $0$ since $x$ is being substituted $a$ prior to the derivative? Or something else?

Thank you.

Answer 1

1. $$\frac{\partial}{\partial x}\left[f(x,y)\vert_{x=a}\right]=\frac{\partial}{\partial x}[f(a,y)]=0$$ since the expression $f(a,y)$ no longer contains $x$ and is thus constant in that respect.

2. $$\left[\frac{\partial}{\partial x}f(x,y)\right]_{x=a}=[f_x(x,y)]_{x=a}=f_x(a,y)$$ where $f_x$ denotes the partial derivative with respect to $x$. The end result can be zero but not necessarily.

Answer 2

The answer for the expression as you have it is zero because you are taking derivative of a function which is not dependent on $x$ with respect to $x$