Mlopez14
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For b), I differentiate y using the quotient rule and simplified a bit:

y = \frac{\frac{\sqrt{x}}{x}- \frac{ln(x)}{2\sqrt{x}}}{x}

Now how do I simplify this completely afterwards?
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_gcx
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Think you meant to write:

\displaystyle \frac {\mathrm dy} {\mathrm dx} = \frac {\frac {\sqrt x} x - \frac {\ln x} {2 \sqrt x}} x.

You don't have to simplify completely because the question isn't asking for that (indeed, you might create extra steps doing that, doesn't here but it's still wasted effort). To find the x-coordinate of M you want to solve \tfrac {\mathrm dy} {\mathrm dx} = 0. That is:

\displaystyle \frac {\mathrm dy} {\mathrm dx} = \frac {\frac {\sqrt x} x - \frac {\ln x} {2 \sqrt x}} x = 0,

multiplying by x:

\displaystyle \frac {\sqrt x} x - \frac {\ln x} {2 \sqrt x} = 0,

at which point you're almost there, you can use index laws to simplify the first term. Then there's a common denominator, which you can multiply through to get something easily solvable.

If you want the full simplification anyway, it helps to write:

\displaystyle \frac {\frac {\sqrt x} x - \frac {\ln x} {2 \sqrt x}} x = \frac 1 x \times \left(\frac {\sqrt x} x - \frac {\ln x} {2 \sqrt x}\right).
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