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f(x) = x² + 3x
The difference quotient is
f(x+h)-f(x)
h
First we find f(x+h) separately:
f(x+h) = (x+h)² + 3(x+h)
f(x+h) = (x+h)(x+h) + 3(x+h)
Multiply the parentheses out and simplify:
f(x+h) = x² + 2hx + h² + 3x + 3h
We put that in place of f(x+h)
and we put (x² + 3x) in place of f(x) in
f(x+h)-f(x)
h
being sure to put parentheses around (x² + 3x) since
it has a - sign before it:
x² + 2hx + h² + 3x + 3h - (x² + 3x)
h
Next we remove those parentheses we just put around
x² + 2hx + h² + 3x + 3h - x² - 3x
h
Notice that all terms cancel except the terms that contain h:
2hx + h² + 3h
h
Next we factor h out of the top:
h(2x + h + 3)
h
Finally we cancel the h's
h(2x + h + 3)
h
2x + h + 3
That's it!
[Sometimes "∆x" is used instead of "h". if your course uses "∆x"
then "∆x" must be treated just as "h" above, and never separated
or considered as "∆" times "x". Ignore this if your book uses h.]
Edwin
