How do I simplify this fractional expression?

Question

$\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}}$ and these numbers are all above the denominator of $h$.

Can someone please help me to understand how to simplify this expression?

Answer 1

I'm not sure if I interpreted your math correctly, but in general if you have something like

$\frac1{\sqrt{a}-\sqrt{b}}$, multiply the numerator and denominator by $\sqrt{a} + \sqrt{b}$

This will cancel out the square root operators from the denominator.

In your case, if I interpreted correctly, multiply the numerator and denominator by $\frac1{\sqrt{x+h}}+\frac1{\sqrt{x}}$.

Answer 2

Combine and then simplify the numerator:

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

Use the fact that

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

to get

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

I imagine you need this to compute the derivative of $1/\sqrt{x}$.

Answer 3

If you knew the derivation, I would tell you that $$\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}\sim h\left(\frac{1}{\sqrt{x}}\right)'=h\frac{1}{-2\sqrt{x^3}}$$ when $h$ is so small.