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!
$\displaystyle f(x) = \frac{2}{\sqrt{x-3}}$
$\displaystyle f(x+h) = \frac{2}{\sqrt{x+h -3}}$
$\displaystyle \lim_{h \to 0} \, \frac{1}{h}\left(\frac{2}{\sqrt{x+h -3}} - \frac{2}{\sqrt{x-3}}\right)$
$\displaystyle \lim_{h \to 0} \, \frac{2}{h}\left(\frac{\sqrt{x-3} - \sqrt{x+h-3}}{\sqrt{x-3} \cdot \sqrt{x+h -3}}\right)$
$\displaystyle \lim_{h \to 0} \, \frac{2}{h}\left(\frac{\sqrt{x-3} - \sqrt{x+h-3}}{\sqrt{x-3} \cdot \sqrt{x+h -3}}\right) \cdot \frac{\sqrt{x-3} + \sqrt{x+h-3}}{\sqrt{x-3} + \sqrt{x+h-3}}$
$\displaystyle \lim_{h \to 0} \, \frac{2}{h}\left(\frac{(x-3) - (x+h-3)}{(x-3)\sqrt{x+h -3}+\sqrt{x-3}(x+h-3)}\right)$
$\displaystyle \lim_{h \to 0} \, \frac{2}{h}\left(\frac{-h}{(x-3)\sqrt{x+h -3}+\sqrt{x-3}(x+h-3)}\right)$
$\displaystyle \lim_{h \to 0} \, -\frac{2}{(x-3)\sqrt{x+h -3}+\sqrt{x-3}(x+h-3)} = -\frac{2}{2(\sqrt{x-3})^3} = -\frac{1}{(x-3)^{\frac{3}{2}}}$
... going to bed.