How do you differentiate arctan(1/x)?

1 Answer
Nov 9, 2016

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

Explanation:

We can write y=arctan(1/x) <=>tany=1/x

:. 1/tany=x
:. coty=x ..... [1]

We can then differentiate implicitly:

-csc^2y dy/dx= 1
dy/dx= -1/csc^2y

Using the identity 1+cot^A-=csc^2A we have

dy/dx= -1/(1+cot^2y)
dy/dx= -1/(1+x^2) (from [1])