Factorising this formula

Question

I came across this simplification in an iTunes U calculus course.

$$\frac{\frac{1}{x_{0}+\delta x}-\frac{1}{x_{0}}}{\delta x} = \frac{1}{\delta x} \left(\frac{x_{0}-(x_{0}+\delta x)}{(x_0+\delta x)x_{0}}\right)$$

I didn't understand how this was done. I can see the denominator being taken out for $\frac{1}{\delta x}$ but do not understand the remainder. Can someone give me some guidance? Thanks

Answer 1

It is $$\frac 1a-\frac 1b=\frac{b-a}{ab}.$$ So, if $a=x_0+\delta x$ and $b=x_0$ you have

$$\frac 1{x_0+\delta x}-\frac 1{x_0}=\frac{x_0-(x_0+\delta x)}{x_0(x_0+\delta x)}.$$

Answer 2

Basically it is taking the same denominator: $$\frac1a-\frac1b=\frac{b}{ab}-\frac{a}{ab}=\frac{a-b}{ab}.$$ In the special case $a=x_0$, $b=x_0+\delta x$