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 […]

The Product Rule

In this post, I derive the so-called product rule that is taught in a Calculus course. The product rule enables one to find the derivative of a function which can be expressed as a product of two functions. That is,  the product rule allows for evaluating $ \frac{d h(x)}{dx} \big|_{a^+}$ with $ h(x) \equiv f(x) g(x) $. To start, use the corresponding definition of a derivative from this post : $ \frac{d h(x)}{dx} \big|_{a^+} = \lim_{\Delta x \rightarrow 0^+} \frac{h(a + \Delta x) – h(a)}{\Delta x}$ Substitute $ h(x) \equiv f(x) g(x) $ : $ \frac{d h(x)}{dx} \big|_{a^+} = \lim_{\Delta x

The Product RuleRead More »