Find the derivative of the function.

Answer

tan(arctan *x*) = *x*.

Taking derivatives of both sides, we find

sec^{2}(arctan *x*)(arctan *x*)' = *x*therefore

Let θ = arctan *x*. Since

somewhere there's a right triangle that looks like this:

The Pythagorean Theorem tells us that the missing side is

sec is the reciprocal of cos, in other words, "hypotenuse over adjacent," therefore

Therefore

In the same way, we can find that

However, these derivatives don't seem to be used nearly as often as the other ones.