What is the derivative of $arctan (1/x)$ ?

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Answer 1 · 2018-06-13T10:09:07

$d/dx arctan(1/x) = -1/(x^2+1)$

We seek:

$d/dx arctan(1/x)$

Using the standard result:

$d/dx tanx=1/(1+x^2)$

In conjunction with the power rule and the chain rule we get:

$d/dx arctan(1/x) = 1/(1+(1/x)^2) \ d/dx (1/x)$

$" " = 1/(1+1/x^2) \ (-1/x^2)$

$" " = -1/((1+1/x^2)x^2)$

$" " = -1/(x^2+1)$

Observation:

The astute reader will notice that:

$d/dx arctan(1/x) = -d/dx arctan x$

From which we conclude that:

$arctan(1/x) = -arctan x + C => arctan(1/x)+arctan x = C$

Although this result may look like an error, it is in fact correct, and a standard trigonometric result:

What is the derivative of arctan (1/x)arctan (1/x)?