Am I using the right technique to differentiate?

[nil101]

Hi
Can you tell me where I'm going wrong with this one? Should I be using the quotient rule, or something else to differentiate y?

\(\displaystyle \L \b y = \frac{x}{{\sqrt x + 1}}\)

\(\displaystyle \L
u = x{\rm }u' = 1\)

\(\displaystyle \L v = \sqrt x + 1{\rm }v' = \frac{1}{2}(x)^{ - {\textstyle{1 \over 2}}}\)

\(\displaystyle \L \frac{{dy}}{{dx}} = \frac{{(\sqrt x + 1) - \frac{1}{2}x(x)^{ - {\textstyle{1 \over 2}}} }}{{(\sqrt x + 1)^2 }}\)

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

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

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

This is not the correct answer. Can you help?

Thanks

 

[galactus]

You're close. The technique is right, just some algebra trouble.

Quotient rule:

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

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

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

 

[nil101]

Thanks for the help Galactus.