If y = tan-1x, then tan y = x. Taking the derivative of the second expression implicitly gives:
solving for the derivative gives:
(1)
This is correct but unsatisfying - we want the derivative in terms of x. Looking at the equation tan y = x geometrically, we get:
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 . From the triangle,
Substituting this into equation (1) above, we get:
last update February 6, 2009 by JL Stanbrough