1. ## Finding the derivative using the definition (have answer)

Find the derivative using the definition (NOT shortcuts, we must use definition).

y=1/sqrt (x)

My friend and I got it down to x^(-1) - x^(-3/2)

How did they get that? (not using shortcuts)
Thanks!

2. $\displaystyle \frac{{\frac{1} {{\sqrt {x + h} }} - \frac{1} {{\sqrt x }}}} {h} = \frac{{\sqrt x - \sqrt {x + h} }} {{h\sqrt x \left( {\sqrt {x + h} } \right)}} = \frac{{ - h}} {{h\sqrt x \left( {\sqrt {x + h} } \right)\left( {\sqrt x + \sqrt {x + h} } \right)}}$

3. $\displaystyle \lim_{\Delta x \to 0} \frac{\frac{1}{\sqrt{x+\Delta x}} - \frac{1}{\sqrt{x}}}{\Delta x}$

$\displaystyle = \lim_{\Delta x \to 0} \frac{\frac{1}{\sqrt{x+\Delta x}} - \frac{1}{\sqrt{x}}}{\Delta x} \cdot \frac{\frac{1}{\sqrt{x+\Delta x}} + \frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x+\Delta x}} + \frac{1}{\sqrt{x}}}$

$\displaystyle = \lim_{\Delta x \to 0} \frac{\frac{1}{x+\Delta x}-\frac{1}{x}}{\Delta x \Big(\frac{1}{\sqrt{x+\Delta x}} + \frac{1}{\sqrt{x}}\Big)}$

$\displaystyle = \lim_{\Delta x \to 0} \frac{\frac{-1}{x(x+ \Delta x)}}{\frac{1}{\sqrt{x+\Delta x}} + \frac{1}{\sqrt{x}}}$

$\displaystyle = \frac{\frac{-1}{x^{2}}}{\frac{2}{\sqrt{x}}} = \frac{-1}{2x^{3/2}}$

4. Originally Posted by Random Variable
$\displaystyle \lim_{\Delta x \to 0} \frac{\frac{1}{\sqrt{x+\Delta x}} - \frac{1}{\sqrt{x}}}{\Delta x}$

$\displaystyle = \lim_{\Delta x \to 0} \frac{\frac{1}{\sqrt{x+\Delta x}} - \frac{1}{\sqrt{x}}}{\Delta x} \cdot \frac{\frac{1}{\sqrt{x+\Delta x}} + \frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x+\Delta x}} + \frac{1}{\sqrt{x}}}$

$\displaystyle = \lim_{\Delta x \to 0} \frac{\frac{1}{x+\Delta x}-\frac{1}{x}}{\Delta x \Big(\frac{1}{\sqrt{x+\Delta x}} + \frac{1}{\sqrt{x}}\Big)}$

$\displaystyle = \lim_{\Delta x \to 0} \frac{\frac{-1}{x(x+ \Delta x)}}{\frac{1}{\sqrt{x+\Delta x}} + \frac{1}{\sqrt{x}}}$

$\displaystyle = \frac{\frac{-1}{x^{2}}}{\frac{2}{\sqrt{x}}} = \frac{-1}{2x^{3/2}}$

THANK YOU!! You're a genius lol. I'm so happy I know how to do this now. :]