# 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: