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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 $\frac{1}{x+h} - \frac{1}{x}$ ?

Gerry Myerson
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addde
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  • Are you doing Calculus? and having trouble simplifying fractions? If so, you are in for a very, very rough ride. – Gerry Myerson Jul 22 '15 at 13:04
  • No i am not doing Calculus but i am doing an math course in the university and i know this is a basic step one proberly should know. – addde Jul 22 '15 at 14:10

2 Answers2

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$f(x+h)-f(x) / h$ 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.

tropianhs
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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.

Lubin
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