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## Homework Statement

f(x) = √x^2 + x + 1

## Homework Equations

Chain Rule

## The Attempt at a Solution

f ' (x) = (x^2+x+1)^1/2

= 1/2 (x^2+x+1)^-1/2(x^2+x+1)'

= 1/2(x^2+x+1)^-1/2(2x + 1)

= 1/2(2x+1)/√(x^2+x+1)

f '' (x) = So I am having trouble with that. Unless my answer for the first derivative is incorrect. I've already tried the quotient rule as well as the product rule. I haven't succeed in any.

Here, I'll use the product rule:

f '' (x) = 1/2(2x+1)(x^2+x+1)^-1/2

= (x+1/2)' (x^2+x+1)^-1/2 + (x+1/2)(x^2+x+1)^-1/2'

= (1)(x^2+x+1)^-1/2 + (x+1/2)(-1/2)(x^2+x+1)^-3/2(x^2+x+1)'

= (x^2+x+1)^-1/2 + (-1/2x - 1/4)(x^2+x+1)^-3/2(2x+1)

= (x^2+x+1)^-1/2 + (-1 -1/2x - 1/2x - 1/4)(x^2+x+1)^-3/2

= (x^2+x+1)^-1/2 + (-4-4x-1/4)(x^2+x+1)^-3/2

= (-5/4 - x)(x^2+x+1)^-3/2 + 1/(x^2+x+1)^1/2

I know for sure this is not the right answer.

The right answer should be 3/4(x^2+x+1)^3/2