Derivative. arctan(x - sqrt(1+x^2))

[smith007]

Homework Statement

Find the derivative of the function. Simplify where possible.
arctan(x - sqrt(1+x²))

Homework Equations

Chain rule

The Attempt at a Solution

let u = x - sqrt(1+x²)

y = tan^-1(u)

dy/du = 1/(1+ u²)
sub in u.
dy/du = 1/1 + (x - sqrt(1+x²))²


du/dx = 1 - 1/2 (1+x²)^-1/2 (2x)

dy/dx = (1/1 + (x - sqrt(1+x²))²) (1 - 1/2 (1+x²)^-1/2 (2x))


I always seem to forget 1 term and I am basically wondering if my method here is correct.

 

[Redbelly98]

Looks right to me. It would be good to include another pair of parentheses in the first term:

dy/dx = (1/(1 + (x - sqrt(1+x²))²)) (1 - 1/2 (1+x2)^-1/2 (2x))

 

[smith007]

Thank you.

I am new to the forums here and I must say they are great!

 

[Redbelly98]

You're welcome, and welcome to PF.