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### Deriving Arctan without Using Numbers

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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?
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```
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

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/
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Associated Topics:
High School Calculus

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