Let's talk math. Specifically, calculating derivatives through differentiation. I've been taking a calculus course at my local university lately, and one of the things that's come up and bit me a couple times is how a lot of the reference material we're given for differentiation shows off differentiation in a way that isn't entirely clear. So in this post I want to record for the future me just how differentiation works.

# Implicit derivatives

When the common derivatives are listed, they're listed for the variable $x$, and the fact that you'll need to apply the chain rule ($(f \circ g)' = (f ' \circ g)g'$ or $(f(g(x)))' = f'(g(x)g'(x)$) when working with expressions is not really made clear. For instance, let's look at the derivative for natural logarithms: $$ (\ln |x|)' = \frac{1}{x} $$

This works great for a single variable $x$, because the derivative of $x$ is $1$. However, in cases where $x$ is an expression ($2x$, $x^2$, $\sin x$, ...), you'll need to multiply that fraction with the derivative of the expression $x$. For this reason, I think it'd be much clearer if the formula sheet had it listed like this: $$ (ln |x|)' = \frac{x'}{x} $$

Similarly, this also goes for other derivatives, like sines, cosines, powers of $e$, and so on:

$$(\arcsin x)' = \frac{x'}{\sqrt{1 - x^2}}$$