Expert: Paul Klarreich - 10/12/2011

Question
Hello,

my name is Ovidiu and I don't know how to derive:
sqrt(1+x^2)

From what I know it should be something like:
1/2*(1+x^2)'/sqrt(1+x^2)=x/sqrt(1+x^2)

But the result should be x*x'/sqrt(1+x^2), and I don`t know why.

Thank you in advance.

Answer
Country:Cluj, Romania
Category:Calculus
Private:No
Subject:derive sqrt of sums of squares
Question:Hello,

my name is Ovidiu and I don't know how to derive:
sqrt(1+x^2)
>>>>>>>>>>> you mean 'differentiate?'

From what I know it should be something like:
1/2*(1+x^2)'/sqrt(1+x^2)=x/sqrt(1+x^2)

>> Yes, that is correct.  I assume you mean here, that you are setting:

y = sqrt(1+x^2)

and finding  dy/dt, not dy/dx.  Now:

dy/dt = dy/dx dx/dt   <<< chain rule

and dy/dx = 1/2*(1+x^2)'/sqrt(1+x^2)=x/sqrt(1+x^2)  << you got this

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

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

NOW do

dy/dt = dy/dx dx/dt

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

and I think you mean  x' as  dx/dt.  That should do it.

But the result should be x*x'/sqrt(1+x^2), and I don`t know why.

Thank you in advance.