The product rule with square roots I am suppose to find the derivative of $H(u) = (u - \sqrt{u})(u - \sqrt{u})$
I know the formula is the derivative of the second function times the first function plus the derivative of the first function times the second function.
I know that it will be $(1-(1/2) u^{-1/2})(1-(1/2) u ^{-1/2})$  I am pretty certain my problem comes from trying to multiply the $(1/2)u^{-1/2}$ term with itself. What exactly happens? I am getting$(1/2)u^{-1/2}$ I am pretty bad at math so I probably made some simple error. All together I get that the derivative is $2u - u^{-1/2} +1$ 

If you are going to use the Product Rule, you have
$$\begin{align*}
H'(u) &= \left(u-\sqrt{u}\right)'\left(u-\sqrt{u}\right) + \left(u-\sqrt{u}\right)\left(u-\sqrt{u}\right)'\\
&= \left(u - u^{1/2}\right)'\left(u-u^{1/2}\right) + \left(u-u^{1/2}\right)\left(u-u^{1/2}\right)'\\
&= \left( 1 - \frac{1}{2}u^{-1/2}\right)\left(u - u^{1/2}\right) + \left(u-u^{1/2}\right)\left(1 - \frac{1}{2}u^{-1/2}\right)\\
&= 2\left(u - u^{1/2}\right)\left(1 - \frac{1}{2}u^{-1/2}\right).
\end{align*}$$
The first step is the Product Rule. The second step is just the fact that $\sqrt{u}=u^{1/2}$. The third step uses the Sum Rule and the Power Rule. The fourth and final step is just the fact that the two summands are equal. 
If you want to further multiply out the product, we have:
$$\begin{align*}
H'(u) &= 2\left(u - u^{1/2}\right)\left(1 - \frac{1}{2}u^{-1/2}\right)\\
&= \left(u - u^{1/2}\right)\left(2 - u^{-1/2}\right)\\
&= 2u - uu^{-1/2} - 2u^{1/2} + u^{1/2}u^{-1/2}\\
&= 2u - u^{1/2} -2u^{1/2}+1\\
&= 2u-3u^{1/2} + 1.
\end{align*}$$

For the function you identify in the comments as the "correct one":
$$H(u) = (u-\sqrt{u})(u+\sqrt{u})$$
assuming you want to exercise the Product Rule, we have:
$$\begin{align*}
H'(u_) &= \left( u - \sqrt{u}\right)'\left(u+\sqrt{u}\right) + \left(u-\sqrt{u}\right)\left(u+\sqrt{u}\right)'\\
&= \left( u - u^{1/2}\right)'\left(u+u^{1/2}\right) + \left(u-u^{1/2}\right)\left(u+u^{1/2}\right)'\\
&=\left(1 - \frac{1}{2}u^{-1/2}\right)\left(u+u^{1/2}\right) + \left(u - u^{1/2}\right)\left(1 + \frac{1}{2}u^{-1/2}\right)\\
&= u+u^{1/2}-\frac{1}{2}u^{-1/2}u - \frac{1}{2}u^{-1/2}u^{1/2} + u + \frac{1}{2}uu^{-1/2} - u^{1/2} - \frac{1}{2}u^{1/2}u^{-1/2}\\
&= u + u^{1/2} - \frac{1}{2}u^{1/2} - \frac{1}{2} + u + \frac{1}{2}u^{1/2} - u^{1/2}-\frac{1}{2}\\
&= 2u-1.
\end{align*}$$
You can verify this is correct, since
$$(u-\sqrt{u})(u+\sqrt{u}) = u^2 - u,$$
and
$$(u^2-u)' = 2u-1.$$
