Question

What is the derivative of `y = arctan(x - sqrt(1+x^2))`?

Answer 1

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

Explanation:

The derivative of `arctanx` is `d/dxarctanx=1/(1+x^2)`, so the chain rule tells us that when we have a function inside the arctangent function, `d/dxarctanu=1/(1+u^2)(du)/dx`.

Thus:

`d/dxarctan(x-sqrt(1+x^2))=1/(1+(x-sqrt(1+x^2))^2)d/dx(x-sqrt(1+x^2))`

Note that `(x-sqrt(1+x^2))^2=x^2-2xsqrt(1+x^2)+(1+x^2)`.

Also note that `d/dxsqrt(1+x^2)=d/dx(1+x^2)^(1/2)=1/2(1+x^2)^(-1/2)(2x)=x/sqrt(1+x^2)`.

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

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

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