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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 $2u - u^{-1/2} +1$

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The function you have as what you know "it will be" is incorrect. – Arturo Magidin Sep 20 '11 at 19:48
I edited my submission, if you think something is wrong just please say. – user138246 Sep 20 '11 at 19:48
So how is it wrong? The derivative of U is 1, and derivative of $u^{-1/2}$ is $1/2 u^{-1/2} isn't it? I need to edit my submission again, I panicked and changed it. Regaurdless, either one will give me the wrong answer. – user138246 Sep 20 '11 at 19:50
@Jordan: As you yourself say in the second paragraph, the derivative of a product is not just the product of the derivatives. But what you are claiming is that the derivative of the product is the product of the derivatives. It's not. – Arturo Magidin Sep 20 '11 at 19:52
@Jordan: For future reference, I would pay special attention to how you write $1/2u$, for example. It is much more clear to write $\frac{1}{2}u$ and $\frac{1}{2u}$. You can right-click on the fractions I just wrote to see the $\TeX$ code. – JavaMan Sep 20 '11 at 19:53
up vote 2 down vote accepted

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 $$(u^2-u)' = 2u-1.$$

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I am really bad at math so I probably need this explained to me like I am five. Anyways I follow the first step, it is the rule written out. The second step is just rewritting the square roots, the fourth step is finding the actually derivatives of the first and last functions, but what is going on in the fourth? It seems like the u from the first one was changed to a 1 but I am not sure why. – user138246 Sep 20 '11 at 19:54
@Jordan: That's a typo. I'll fix it. – Arturo Magidin Sep 20 '11 at 19:57
I might have made an error in something, the answer should be 2u-1. – user138246 Sep 20 '11 at 20:02
@Jordan: Is the original problem correct? Could it be perhaps that you were supposed to do the derivative of$$H(u) = \left( u - \sqrt{u}\right)\left(u+\sqrt{u}\right)\ ?$$Note the $+$ sign in one of the factors. – Arturo Magidin Sep 20 '11 at 20:04
I might be dyslexic. I was suppose to do that and I have been doing what I submitted for quite a while. The problem is suppose to be $ H(u) = \left( u - \sqrt{u}\right)\left(u+\sqrt{u}\right)\ $ – user138246 Sep 20 '11 at 20:07

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