Ask Me Help Desk - Finding the derivative of 1/square root of X with definition

Try this (for ease of typing I'm leaving the limit symbol):

Multiply through by $\sqrt x \sqrt {x+h}$ :

$ \frac { \frac 1 {sqrt{ x+h}} - \frac 1 {\sqrt x}} h \times \frac {\sqrt x \sqrt {x+h} } {\sqrt x \sqrt {x+h}} = \frac { \sqrt x - \sqrt{x+h}} {h \sqrt x \sqrt {x+h} } $

Whenever you have the difference of square roots like this it helps to multiply the numerator and denominator by the sum of those roots, and see where it takes you:

$ \frac { \sqrt x - \sqrt{x+h}} {h \sqrt x \sqrt {x+h} } = \frac { \sqrt x - \sqrt{x+h}} {h \sqrt x \sqrt {x+h} } \cdot \frac { \ sqrt x + \sqrt{x+h}} { \ sqrt x + \sqrt{x+h}} = \frac {x - (x+h)} {h \sqrt x \sqrt{x+h} (\sqrt x + \sqrt {x+h})} = \frac 1 {\sqrt x \sqrt {x+h} (\sqrt x + \sqrt {x+h})} $

Now take the limit as h goes to 0.