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

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Answer 1 · 2015-07-31T13:07:54

The derivative is: $(-1)/(x^2+1)$

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

So

$d/dx arctan(u) = 1/(1+u^2) (du)/dx$

And

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

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

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

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

Faster Method?

Use the fact that $arctan(1/x) = arc cot(x)$

and

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

to go straight to the answer.

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