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} […]

What is $ \lim_{x\rightarrow0} \frac{\sin x}{x}$ ? Recall the definition of a limit, repeated here for reference [2]. A function $f(x)$ approaches a limit $A$ as $x$ approaches $a$ if, and only if, for each positive number $\epsilon$ there is another, $\delta$, such that whenever $0 < |x-a| < \delta $ we have $|f(x) – A| < \epsilon$. That is, when $x$ is near $a$ (within a distance $\delta$ from it), $f(x)$ is near $A$ (within a distance $\epsilon$ from it). In symbols we write $\lim_{x \rightarrow a} f(x) = A$. Using this definition, $a$ is $0$ in this case. Now

The sine function divided by its angle, and a certain limitRead More »

The following drawing shows how to convert from Cartesian coordinates to spherical coordinates. This is a short post, but these three equations are pretty useful. I am going to use the end of this post to define the cosine and sine functions: $ \sin\phi \equiv \frac{opp}{hyp}$ $ \cos\phi \equiv \frac{adj}{hyp}$. Here, $ hyp$ is the length of the hypotenuse of a right triangle, $ opp$ is the length of the side that is opposite to the angle $ \phi$, and $ adj$ is the length of the side that is adjacent to the angle $ \phi$. Note that $

Spherical CoordinatesRead More »