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

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Answer 1 · 2016-06-20T22:39:06

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

#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#