# What can be said about differentiation of a constant with respect to a constant & differentiation of a variable with respect to a constant?

Dec 21, 2017

#### Explanation:

One of the definition of the derivative of function $f$ with respect to function $g$ is

$\frac{\mathrm{df}}{\mathrm{dg}} = {\lim}_{h \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{g \left(x + h\right) - g \left(x\right)}$

If $g \left(x\right)$ is a constant function, then $g \left(x + h\right) = g \left(x\right)$ for all $x$ and all $h$, so the denominator is always $0$ and no limit exists.

This is true regardless of whether $f \left(x\right)$ is constant or non-constant.

Another definition

$\frac{\mathrm{df}}{\mathrm{dg}} = \frac{\mathrm{df}}{\mathrm{dx}} \cdot \frac{1}{\frac{\mathrm{dg}}{\mathrm{dx}}}$

But if $g$ is constant then $\frac{\mathrm{dg}}{\mathrm{dx}} = 0$, so the second factor is not defined.
Again whether $f$ is constant or variable, no derivative exists.