Derivative with chain rule

[MarkSA]

I have the problem:

Find the derivative of:
sqrt(x^2 + x)

Which of these methods is correct?
a. = 1/2[(x^2 + x)^(-1/2)] * (2x + 1)

Or
b. = 1/2[(x^2 + x)^(-1/2)] * (2x + 1) * (2)

From messing with the graphing calculator, it looks like the answer is the first one (a.) Can you explain why a person would stop at that point on the problem instead of continuing and finding the derivative of (2x + 1) by chain rule?

 

[MarkSA]

I think i'm seeing now..

Since I used the power rule on
x^2 + x
to get 2(x) + 1, x is what the chain rule would have been used on, not 2x+1. Is this correct?

 

[galactus]

\(\displaystyle \L\\\sqrt{x^{2}+x}\)

Chain rule is basically 'derivative of outside times derivative of inside'.

Inside: \(\displaystyle \L\\2x+1\)

Outside: \(\displaystyle \L\\\frac{1}{2}(x^{2}+x)^{\frac{-1}{2}}\)

So, you have \(\displaystyle \L\\\frac{1}{2}(x^{2}+x)^{\frac{-1}{2}}\cdot{(2x+1)}\)

=\(\displaystyle \L\\\frac{2x+1}{2\sqrt{x^{2}+x}}\)