Deriving Arctan without Using NumbersDate: 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/ |
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