Cut Down Derivative Table Memorization in Half

This is a useful rule that you won’t find in most books for calculating the derivative of an inverse function. This includes any of the “arc” functions in trigonometry for example.

Let’s start with a function f(x). Let’s say that its derivative is f’(x).

Now let h(x) be the inverse function of f(x). That is, f(h(x))=h(f(x))=x. If we graph f and h, we see that h is a reflection of f across the line y=x.

Now let’s calculate h’(x).

Let’s assign the variable y=h(x). Then we can work out the derivation rule for inverse functions.

Let’s try this on a couple examples.

We know that:

Now let’s apply this rule on the inverse of f.

We also know that:

Now let’s apply this rule on the inverse of f.

We can simplify this further using the trigonometric identity

Just set

Next time I will show you the general rule for integrating an inverse function.