Find the derivative of the arctangent of x function

y = arctan(x)Start by taking the tangent of both sides:tan(y) = xTake the derivative of each side with respect to x, using implicit differentiation/the chain rule for the LHS, then rearrange to make dy/dx the subject:dy/dx = 1/sec^2(y)Use sec^2(y) = 1 + tan^2(y) to change the denominator:dy/dx = 1/(1 + tan^2(y))Plugging our original definition of y into this we get our final result:dy/dx = 1/(1 + tan^2(arctan(x))) = 1/(1 + x^2)