## The Derivative of a Constant Function

From this post, one definition of a derivative is

$\lim_{\Delta x\rightarrow0^+}\frac{f(a+\Delta x)-f(a)}{\Delta x}\equiv\frac{d f(x)}{dx}\big|_{a^+}$.

In this case, the values of $\Delta x$ are restricted to positive values due to the $+$ in $0^+$ written in the limit.

A function that does not vary with respect to an independent variable is called a constant function. On a graph with perpendicular $x$ and $y$ axes, a constant function looks like a horizontal line. The slope, or $\frac{\Delta y}{\Delta x}$, of a constant function $f(x) = C \in \mathbb{R}$ is equal to $0$ because $\Delta y=0$ and $\Delta x \ne 0$. A slope of a function can be geometrically represented with a line. A line with a slope of zero is a horizontal line with respect to the $x$-axis, since the change in the $y$-coordinate is zero and the change in the $x$-coordinate is nonzero. Conversely, a horizontal line with no change in the $y$-coordinate and a nonzero change in the $x$-coordinate *has a slope of $0$*. The derivative is best understood geometrically in terms of a tangent line. Geometrically, it is self evident that the line tangent to each point of a constant function $f(x)=C \in \mathbb{R}$ is a horizontal line. Hence the tangent line to each point of the constant function has a slope of $0$. This conclusion may be expressed as

$\frac{d f(x)}{dx}\big|_{a^+} = \frac{d C}{dx}\big|_{a^+} = 0$.