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Derivative of the inverse tangent

The first derivative of the inverse tangent is

 

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Explanation

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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