Question 1094525
{{{(f(x+h)-f(x))/h}}}=[f(x+h)-f(x)]/h is the average rate of change of the function between {{{x}}} and {{{x+h}}} .


For {{{f(x)= -2/x+3}}} , with {{{h<>0}}} , it is
{{{(f(x+h)-f(x))/h}}}{{{"="}}}{{{(-2/(x+h)+3-(-2/x+3))/h}}}{{{"="}}}{{{(-2/(x+h)+3+2/x-3)/h}}}{{{"="}}}{{{(-2/(x+h)+2/x)/h}}}{{{"="}}}{{{(-2x/(x(x+h))+2(x+h)/(x(x+h)))/h}}}{{{"="}}}{{{((-2x+2x+2h)/(x(x+h)))/h}}}{{{"="}}}{{{2h/(x(x+h)h)}}}{{{"="}}}{{{highlight(2/(x(x+h)))}}}{{{"="}}}{{{highlight(2/(x^2+xh))}}}
For this problem, the meaning of "simplest" is whatever the teacher thinks it is.
As the expression {{{(f(x+h)-f(x))/h}}} is used to introduce the concept of derivative,
I would expect that {{{(f(x+h)-f(x))/h2/(x^2+xh)}}} is what the teacher had in mind.
 
NOTE: When you type expressions into a computer or calculator,
you should make sure you type the parentheses that were implied in those long fraction bars we can write on paper:
{{{(f(x+h)-f(x))/h}}}=[f(x+h)-f(x)]/h , but f(x+h)-f(x)/h={{{f(x+h)-f(x)/h}}} ,
-2/x+3={{{-2/x+3}}} , and {{{-2/(x+3)}}}=-2/(x+3)
Forgetting to type some of those parentheses implied by other grouping symbols may lead to wrong results.
If you type "0.732-0.002/0.713-0.003" into a calculator, you are likely to get
0.732-0.002/0.713-0.003={{{0.732-0.002/0.713-0.003}}} , and that is approximately {{{0.732-0.002805049-0.03=approximately0.7261974951}}} ,
because most calculators know about order of operations.
If what you meant was
(0.732-0.002)/(0.713-0.003)={{{(0.732-0.002)/(0.713-0.003)=0.730/0.710=approximately1.028169014}}} ,
you got the wrong result.