What is the derivative of #arctan (1/x)#?
1 Answer
Jun 13, 2018
# d/dx arctan(1/x) = -1/(x^2+1) #
Explanation:
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:
