Deriving Arctan without Using Numbers

Date: 03/22/2002 at 02:56:15
From: Matt Barrett
Subject: Arctan

How do you do you derive arctan without using numbers, or how do you 
prove (arctan)' = 1/1+x^2?

Date: 03/22/2002 at 09:05:20
From: Doctor Rick
Subject: Re: Arctan

Hi, Matt.

Are you asking how we find the derivative of arctan(x) with respect to 
x? We can start with the equation

  y = arctan(x)

and note that this is equivalent to

  x = tan(y)

Then take the derivative of each side with respect to x:

  1 = sec^2(y)*dy/dx

Is this enough, or do we also need to prove the derivative of tan(y)? 
This can be done easily by writing tan(y) as sin(y)/cos(y) and using 
the rule for the derivative of a quotient.

Now we solve for dy/dx:

  dy/dx = 1/sec^2(y)

How is sec(y) related to x = tan(y)? We know the identity

  tan^2(y) + 1 = sec^2(y)

(This can be proved easily by writing in terms of sin and cos and 
using the Pythagorean identity.) Thus

  1/sec^2(y) = 1/(x^2 + 1)

This is thus the derivative we seek:

  y = arctan(x)
  dy/dx = 1/(x^2 + 1)

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/