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Differentiation

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 …