How do you differentiate y= arctan(x - sqrt(1+x^2))?

1 Answer
Jun 20, 2016

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

Explanation:

d/dx tan^-1(x) = 1/(1+x^2)

Now, treat x-sqrt(1+x^2) as x in the above definition.

That would give us,
d/dx tan^-1(x-sqrt(1+x^2)) = 1/(1+(x-sqrt(1+x^2))^2

Don't forget the chain rule though!

The derivative of x-sqrt(1+x^2) is 1-(x/(sqrt(1+x^2)))

Multiplying the derivative would give us,

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