Prove that the result of differentiating arc tan (1-x^2) + arc cot (1-x^2) is 0.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

We have to prove that the derivative of arc tan (1-x^2) + arc cot (1-x^2) is 0.

 

We know that the derivative of arc tan x = 1/[1 + (1-x^2)^2] and the derivative of arc cot x = -1 / [1 + (1-x^2)^2]

Let f(x) = arc tan (1-x^2) + arc cot (1-x^2)

f'(x) = -2x * 1/[1 + (1-x^2)^2] + (-2x)* -1/[1 + (1-x^2)^2]

=> -2x/[1 + (1-x^2)^2] + -(-2x)/[1 + (1-x^2)^2]

=> -2x/[1 + (1-x^2)^2] + 2x/[1 + (1-x^2)^2]

=> 0

The derivative of arc tan (1-x^2) + arc cot (1-x^2) = 0.

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial