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 […]

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, use this post to obtain the equations relating Cartesian coordinates to spherical coordinates. In particular: $x = r \sin\theta \cos \phi $ $y = r \sin \theta \sin \phi $ $z = r \cos\theta $ Note that $x$ changes if $r$ changes, $\theta$ changes, and/or $\phi$

The Crux of CalculusRead More »

From this post, one type of derivative is $\lim_{\Delta x\rightarrow0^+}\frac{f(a+\Delta x)-f(a)}{\Delta x}\equiv\frac{df(x)}{dx}\big|_{a^+}$ To be consistent with my previous interpretation of $0^+$ in this post, $\Delta x \rightarrow 0^+$ means constraining $\Delta x$ to positive numbers. Next, define $x$ and $a$ as variables for nonnegative real numbers, to avoid having a negative angle for the sine function. In this post, I find the derivative of $\sin x$ using the previous definition of a derivative. I also approximately follow the steps in reference [1]. Using $f(x) = \sin x$, the derivative is $ \frac{d \sin(x)}{dx} \big|_{a^+} = \lim_{\Delta x \rightarrow 0^+} \frac{\sin(a+\Delta x) – \sin(a)}{\Delta x}

Differentiating the Sine FunctionRead More »

I copy the definitions of three different types of derivatives from [1]: $ \lim_{\Delta x \rightarrow 0} \frac{f(a+\Delta x) – f(a)}{\Delta x} \equiv \frac{d f(x)}{dx}\big|_{a}$ $ \lim_{\Delta x \rightarrow 0^+} \frac{f(a+\Delta x) – f(a)}{\Delta x} \equiv \frac{d f(x)}{dx}\big|_{a^+}$ $ \lim_{\Delta x \rightarrow 0^-} \frac{f(a+\Delta x) – f(a)}{\Delta x} \equiv \frac{d f(x)}{dx}\big|_{a^-}$ These definitions are best understood geometrically in terms of secant and tangent lines. Note that $ \Delta x$ never equals zero because otherwise the fraction would be undefined. The tangent line intersects the function at only one point. [1] David V. Widder. Advanced Calculus. Dover 1989.