Differentiate arctan(x) with respect to x

We can set arctan(x) = y. Remember that arctan(x) is the inverse of tan(x), so we can take the tangent of both sides to give: tan(arctan(x)) = x = tan(y). Tan(x) has the standard derivative of sec^2(x) (which you can derive from the fact that tan(x) = sin(x)/cos(x) and use the quotient rule to differentiate from there), so we can now differentiate both sides with respect to y: x = tan(y), so dx/dy = sec^2(y). Don't be put off by the fact that we are differentiating with respect to y! The same rules apply, we have only changed the "name" of our variable. Using the identity tan^2(y) + 1 = sec^2(y), we can rewrite our expression as dx/dy = tan^2(y) + 1.But remember that we set tan(y) = x, so dx/dy = x^2 + 1. We were asked in the question to find dy/dx, not dx/dy, but that is just 1/(dx/dy) = 1/(x^2 + 1). Hence, the derivative of arctan(x) = 1/(x^2 + 1)