The Derivative of Arctan x

If y = tan-1x, then tan y = x. Taking the derivative of the second expression implicitly gives:

sec^2y dy/dx = 1

solving for the derivative gives:

dy/dx=cos^2y (1)

This is correct but unsatisfying - we want the derivative in terms of x. Looking at the equation tan y = x geometrically, we get:

right triangle where tan y = x

In this right triangle, the tangent of (angle) y is x/1 (opposite/adjacent). Using the Pythagorean Theorem, the length of the hypotenuse is then sqrt(1 + x^2). From the triangle,

Substituting this into equation (1) above, we get:

last update February 6, 2009 by JL Stanbrough