What is the derivative of arctan(1/x)?

chatterbsau3e

chatterbsau3e

Answered question

2023-03-24

What is the derivative of arctan ( 1 x ) ?

Answer & Explanation

Spotty7b2o

Spotty7b2o

Beginner2023-03-25Added 3 answers

The derivative is: - 1 x 2 + 1
Solution:
d d x arctan ( x ) = 1 1 + x 2
Therefore
d d x arctan ( u ) = 1 1 + u 2 d u d x
So
d d x arctan ( 1 x ) = 1 1 + ( 1 x ) 2 d d x ( 1 x )
= 1 1 + 1 x 2 - 1 x 2
= x 2 x 2 + 1 - 1 x 2
= - 1 x 2 + 1
Another Method
Use the fact that arctan ( 1 x ) = a r c cot ( x )
and
d d x a r c cot ( x ) = - 1 1 + x 2
to go straight to the answer.

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