I kept having problems trying to differentiate the following equation with respect to h:

$\displaystyle

\frac{d}{dh} (\frac{100}{h} + 36h(\frac{5}{h})^{1/2})

$

What I did was I tried to split it into two:

$\displaystyle

\frac{d}{dh} \frac{100}{h} + \frac{d}{dh}\left(36h(\frac{5}{h})^{1/2}\right)

$

then..

$\displaystyle

\frac{-100}{h^2} + \frac{d}{dh}\left(36h(\frac{5}{h})^{1/2}\right)

$

from here, I use product rule on the second term:

$\displaystyle

\frac{d}{dh}\left(36h(\frac{5}{h})^{1/2}\right)

$

$\displaystyle

= \left[\frac{d}{dh}(36h)\right] (\frac{5}{h})^{\frac{1}{2}} + \left[\frac{d}{dh} (\frac{5}{h})^{\frac{1}{2}}\right] (36h)

$

$\displaystyle

= 36(\frac{5}{h})^{\frac{1}{2}} + \frac{1}{2}(36h)(\frac{-5}{h^2})(\frac{h}{5})^{\frac{1}{2}}

$

$\displaystyle

= 36(\frac{5}{h})^{\frac{1}{2}} - \frac{90}{h}(\frac{h}{5})^{\frac{1}{2}}

$

the final result of the differentiated equation that I got is..

$\displaystyle

\frac{-100}{h^2} + 36(\frac{5}{h})^{\frac{1}{2}} - \frac{90}{h}(\frac{h}{5})^{\frac{1}{2}}

$

which is

**wrong**!

And I don't know what is wrong and how I should do it.