How to evolve an expression with two denominators

Question

The task is to simplify the expression: $\displaystyle\frac{f(x+h)-f(x)}{h}$ when $\displaystyle f(x) =\frac{1}{x}$.

I don't know how to do this since I get to the step $\displaystyle\frac{\frac{1}{x+h} - \frac{1}{x}}{h}$ which gives me two denominators.

Should I multiply each term with $h$ to cancel out the $h$ at the bottom or should I try to multiply so I get the same denominator in the expression $\frac{1}{x+h} - \frac{1}{x}$ ?

Answer 1

First you make sure that you’ve made the correct substitution. This would be $$ \frac{f(x+h)-f(x)}h = \frac{\frac1{x+h}-\frac1x}{h}\,. $$ There are many ways of handling this, but I recommend scanning the whole expression and multiplying top and bottom of the big fraction by something that removes the fractional nature of the top and the bottom both. In this case, multiply both by $x(x+h)$, and don’t forget to distribute correctly. When you do the multiplication, you get $$ \frac{x-(x+h)}{x(x+h)h}\,. $$ Now collapse and simplify.

Answer 2

$f(x+h)-f(x) / h$ when $f(x) = 1/x$ should become $$\Bigg(\frac{1}{(x+h)} - \frac{1}{x}\Bigg) \frac{1}{h} = \Bigg( \frac{x-x-h}{x(x+h)}\Bigg) \frac{1}{h}$$

You can go on from here and simplify more.