`y = tan^-1(x - sqrt(1 + x^2))` Find the derivative of the function. Simplify where possible.

Textbook Question

Chapter 3, 3.5 - Problem 54 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`y=tan^-1(x-sqrt(1+x^2))`

`d/(dt) tan^-1t=1/(1+t^2)`

`y'=1/(1+(x-sqrt(1+x^2))^2) * d/(dx)(x-sqrt(1+x^2))`

`y'=1/(1+x^2+1+x^2-2xsqrt(1+x^2)) *(1-(1/2)(1+x^2)^(-1/2)(2x))`

`y'=1/(2+2x^2-2xsqrt(1+x^2)) *(1-x/(sqrt(1+x^2)))`

`y'=(sqrt(1+x^2)-x)/(2sqrt(1+x^2)(1+x^2-xsqrt(1+x^2)))`

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