Find the derivative of the function.
tan(arctan x) = x.
Taking derivatives of both sides, we find
sec2(arctan x)(arctan x)' = xtherefore
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
In the same way, we can find that
However, these derivatives don't seem to be used nearly as often as the other ones.