If y = tan^{-1}x, 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:

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last update February 6, 2009 by JL Stanbrough*