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

Question

70615 views around the world

You can reuse this answer
Creative Commons License

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.