Help with a derivative and solving for Critical points

I have the equation

(x^2)/(sqrt(x+1))

This one has been stumping me, not sure how to reduce the derivative properly much less solve for 0.

I get it down to this using the quotient rule:

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

Just started learning derivative a few weeks ago and they're still stumping me. I'm really terrible at simplifying these down. Once it's simplified down how would I solve for ZERO to find the crit points?

Any help would be appreciated.

SammyS
Staff Emeritus
Homework Helper
Gold Member
I have the equation

(x^2)/(sqrt(x+1))

This one has been stumping me, not sure how to reduce the derivative properly much less solve for 0.

I get it down to this using the quotient rule:

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

Just started learning derivative a few weeks ago and they're still stumping me. I'm really terrible at simplifying these down. Once it's simplified down how would I solve for ZERO to find the crit points?

Any help would be appreciated.
Hello LLofantaine. Welcome to PF !

The derivative is correct. The rest is just algebra.

((x+1)^(1/2)*2x - x^2*1/2(x+1)^(-1/2)) / (x+1) → $\displaystyle \frac{2x\sqrt{x+1}-\displaystyle \frac{x^2}{2\sqrt{x+1}}}{x+1}$

Thanks, so that's what the derivative looks like, is there anyway to simplify that down further? Also, my algebra skills are obviously horrible so I can't remember the first thing about setting and solving that for 0

SammyS
Staff Emeritus
Homework Helper
Gold Member
Thanks, so that's what the derivative looks like, is there anyway to simplify that down further? Also, my algebra skills are obviously horrible so I can't remember the first thing about setting and solving that for 0
To reduce the fraction, multiply the "main" numerator & denominator by √(x+1) . That should allow you to simplify the numerator greatly.

(Yes. When students say they're having trouble doing the calculus, it's often the algebra skills that are lacking.)

HallsofIvy
The simplest way to handle this is NOT to use the quotient rule but, instead, write it as $$x^2(x+ 1)^{-1/2}$$ and use the product rule.