First Principles- Calculus

**zachattack** · August 30th 2009, 02:33 PM

Hello math guys, I need desperate help with this first principles problem:

2/sqrt(x-3)

showing how to work through this problem would be great to help me get better at these.

thanks!

**skeeter** · August 30th 2009, 03:22 PM

Originally Posted by zachattack

Hello math guys, I need desperate help with this first principles problem:

2/sqrt(x-3)

showing how to work through this problem would be great to help me get better at these.

thanks!

$f(x) = \frac{2}{\sqrt{x-3}}$

$f(x+h) = \frac{2}{\sqrt{x+h -3}}$

$\lim_{h \to 0} \, \frac{1}{h}\left(\frac{2}{\sqrt{x+h -3}} - \frac{2}{\sqrt{x-3}}\right)$

$\lim_{h \to 0} \, \frac{2}{h}\left(\frac{\sqrt{x-3} - \sqrt{x+h-3}}{\sqrt{x-3} \cdot \sqrt{x+h -3}}\right)$

$\lim_{h \to 0} \, \frac{2}{h}\left(\frac{\sqrt{x-3} - \sqrt{x+h-3}}{\sqrt{x-3} \cdot \sqrt{x+h -3}}\right) \cdot \frac{\sqrt{x-3} + \sqrt{x+h-3}}{\sqrt{x-3} + \sqrt{x+h-3}}$

$\lim_{h \to 0} \, \frac{2}{h}\left(\frac{(x-3) - (x+h-3)}{(x-3)\sqrt{x+h -3}+\sqrt{x-3}(x+h-3)}\right)$

$\lim_{h \to 0} \, \frac{2}{h}\left(\frac{-h}{(x-3)\sqrt{x+h -3}+\sqrt{x-3}(x+h-3)}\right)$

$\lim_{h \to 0} \, -\frac{2}{(x-3)\sqrt{x+h -3}+\sqrt{x-3}(x+h-3)} = -\frac{2}{2(\sqrt{x-3})^3} = -\frac{1}{(x-3)^{\frac{3}{2}}}$

... going to bed.

**zachattack** · August 30th 2009, 04:04 PM

thank you, but can someone please explain how you get 1/h factored out of the rest of the function, I understand the factoring out of the two though

**skeeter** · August 30th 2009, 04:08 PM

$\frac{f(x+h) - f(x)}{h} = \frac{1}{h}[f(x+h) - f(x)]$

**eXist** · August 30th 2009, 04:10 PM

He applied the formula:

$\frac{f(x + h) - f(x)}{h}$

Beat me to it skeets

Go to bed already

.

**zachattack** · August 30th 2009, 04:12 PM

thanks

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