# On differentiation

The muddy partsLet’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 (2*x*, *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:

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