Differentiate arctan(x) with respect to x. Leave your answer in terms of x

Let y = arctan(x). Arctan(x) is difficult to differentiate but I know how to differentiate tan(x) (=sec^2(x)) so take the tan of both sides: tan(y) = x. The next step will be to differentiate both sides. To differentiate tan(y) with respect to x, we can use the chain rule as you have a function of a function. This means that we take the differential of the outer function and multiply it by the differential of the inside function. In this case, we get sec^2(y)*(dy/dx). The differential of the right is just 1 as d(x)/dx is just 1. Hence we get (dy/dx)*sec^2(y) = 1. Rearranging, we get (dy/dx) = 1/sec^2(y). This is a solution but it is not in terms of x so we are halfway there. What we do know however, is that y = arctan(x) so if we can somehow turn the sec^2(y) into something that has tan(y)'s instead, then the tan will cancel with the arctan to produce something with x’s in it, as we want . We can use the identity sec^2(t) = 1 + tan^2(t) to obtain: dy/dx = 1/(1 + tan^2(y)). Finally, using y = arctan(x) in tan(y), we get dy/dx = 1/(1+tan^2(arctan(x)) thus the final answer is: dy/dx= 1/(1+x^2)