I would like to evaluate $ \frac{dx(r)}{dr} \bigg|_{x^+}$ with $x(r) = r \sin\theta \cos\phi$. Substituting, the expression to evaluate is $ \frac{d(r \sin\theta \cos\phi )}{dr} \bigg|_{x^+}$ From the product rule, $ \frac{dx(r)}{dr} \bigg|_{r^+} = \sin\theta \cos\phi \frac{dr}{dr} \bigg|_{r^+} + r \frac{d ( \sin\theta \cos\phi)}{dr} \bigg|_{r^+}$. Each $x$ in the product rule has been replaced by $r$….

# Category: 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…

## 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)…

## The Crux of Calculus

Define $\Delta x \equiv x_2 – x_1$, to be consistent with this post. Similarly, define $\Delta y \equiv y_2 – y_1$ and $\Delta z \equiv z_2 – z_1$. The Cartesian coordinates are $x$, $y$, & $z$. In contrast, the spherical coordinates are $r$, $\theta$, & $\phi$. Here, $\phi$ is the azimuthal angle in the $xy$-plane. Next,…