How do I differentiate this equation?

**xEnOn** · November 7th 2010, 01:46 AM

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

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

What I did was I tried to split it into two:
$ \frac{d}{dh} \frac{100}{h} + \frac{d}{dx}36h(\frac{5}{h})^{1/2} $

then..

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

from here, I use product rule on the second term:
$ \frac{d}{dh}36h(\frac{5}{h})^{1/2} $
$ = \frac{d}{dh}(36h) (\frac{5}{h})^{\frac{1}{2}} + \frac{d}{dx} (\frac{5}{h})^{\frac{1}{2}} (36h) $
$ = 36(\frac{5}{h})^{\frac{1}{2}} + \frac{1}{2}(36h)(\frac{-5}{h^2})(\frac{h}{5})^{\frac{1}{2}} $
$ = 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..
$ \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.

**Archie Meade** · November 7th 2010, 04:34 AM

Originally Posted by xEnOn

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

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

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

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

then..

$ \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:

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

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

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

$ = 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..

$ \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.

$\displaystyle\left(\frac{5}{h}\right)^{\frac{1}{2} }=\frac{\sqrt{5}}{\sqrt{h}}$

$\displaystyle\sqrt{\frac{5}{h}}\frac{d}{dh}36h+36h \frac{d}{dh}\left(\frac{5}{h}\right)^{\frac{1}{2}} =36\sqrt{\frac{5}{h}}+36h\sqrt{5}\frac{d}{dh}h^{-\frac{1}{2}}$

$=\displaystyle\ 36\left(\frac{5}{h}\right)^{\frac{1}{2}}+36h(5)^{\ frac{1}{2}}\left[-\frac{1}{2}h^{-\frac{3}{2}}\right]=36\left(\frac{5}{h}\right)^{\frac{1}{2}}-\frac{36h(5)^{\frac{1}{2}}}{2h^{\frac{3}{2}}}$

$=\displaystyle\ 36\left(\frac{5}{h}\right)^{\frac{1}{2}}-18\left(\frac{5}{h}\right)^{\frac{1}{2}}=18\left(\ frac{5}{h}\right)^{\frac{1}{2}}$

which is what you have, if you simplify...

$\displaystyle\ -\frac{90}{h}\frac{\sqrt{h}}{\sqrt{5}}=-\frac{18(5)}{\sqrt{h}\sqrt{h}}\left(\frac{\sqrt{h} }{\sqrt{5}}\right)=-\frac{18\sqrt{5}\sqrt{5}\sqrt{h}}{\sqrt{5}\sqrt{h} \sqrt{h}}=-\frac{18\sqrt{5}}{\sqrt{h}}$

To make things easier, you could write

$\displaystyle\ 36h\frac{\sqrt{5}}{\sqrt{h}}=36\sqrt{5}\sqrt{h}$

Then

$\displaystyle\ 36\sqrt{5}\frac{d}{dh}h^{\frac{1}{2}}=36\sqrt{5}\f rac{1}{2}h^{-\frac{1}{2}}=18\frac{\sqrt{5}}{\sqrt{h}}$

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