# On differentiation

The muddy parts

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 ∘ g)′ = (f′ ∘ 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, x2, 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:

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