What is the derivative of a a variable divided by its mean w.r.t. the variable?

Question

For a variable $x$, whose mean is given by $\overline{x}$, what is:

$\frac{\partial}{\partial x} \frac{x}{\overline{x}}$?

I think it is $1/x$ but cannot prove it.

Answer 1

The mean of a variable x is independent of x. In other words, $\bar{x}$ is a constant with respect to x.

The derivative of a variable with respect to itself is 1. In other words, $\frac{\partial}{\partial x}x=1$.

Putting those together:

$$\frac{\partial}{\partial x}\frac{x}{\bar{x}} = \frac{1}{\bar{x}}\frac{\partial}{\partial x}x = \frac{1}{\bar{x}}$$