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Derivative of the inverse tangent

The first derivative of the inverse tangent is



We'll start with the tangent, for which applies

x = tan y         ⇔         y = tan−1 x

Differentiation gives

Turning over delivers

We want to trace this back to a function that only contains tan y, because we can then replace that with x. With the fundamental formula of trigonometry we get

Division of the numerator and denominator by cos2y shows

Substitution of y = tan−1 x and tan y = x gives

so that


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