Hello, ryansmith069!
The most common problem is simply not understanding the Difference Quotient.
If \(\displaystyle f(x)\:=\:4x^2\,+\,5x\,-\,5\), then \(\displaystyle \frac{f(x + h) - f(x)}{h}\:= ?\)
That messy expression is actually a "recipe" . . . just follow its instructions.
. . [1] Find \(\displaystyle f(x\,+\,h)\) . . . Replace \(\displaystyle x\) with \(\displaystyle x+h\) ... and simplify.
. . [2] Subtract \(\displaystyle f(x)\) . . . Subtract the original function ... and simplify.
. . [3] Divide by \(\displaystyle h\) . . . . . Usually means "factor and cancel".
Here we go . . .
\(\displaystyle [1]\;\;f(x\,+\,h)\:=\:4(x\,+\,h)^2\,+\,5(x\,+\,h)\,-\,5\)
. . . . . . . . . . . . .\(\displaystyle =\:4(x^2\,+\,2xh\,+\,h^2)\,+\,5(x\,+\,h)\,-\,5\)
. . . . . . . . . . . . .\(\displaystyle =\:4x^2\,+\,8xh\,+\,4h^2\,+\,5x\,+\,5h\,-\,5\)
\(\displaystyle [2]\;\;f(x\,+\,h)\,-\,f(x)\:=\
4x^2\,+\,8xh\,+\,4h^2\,+\,5x\,+\,5h\,-\,5)\,-\,(4x^2\,+\,5x\,-\,5)\)
. . . . . . . . . . . . . . . . . . .\(\displaystyle =\;4x^2\,+\,8xh\,+\,4h^2\,+\,5x\,+\,5h\,-\,5\,-\,4x^2\,-\,5x\,+\,5\)
. . . . . . . . . . . . . . . . . . .\(\displaystyle =\:8xh\,+\,4h^2\,+\,5h\)
**
\(\displaystyle [3]\;\;\frac{f(x\,+\,h)\,-\,f(x)}{h}\;=\;\frac{8xh\,+\,4h^2\,+\,5h}{h}\;=\;\frac{h(8x\,+\,4h\,+\,5)}{h}\:=\:8x\,+\,4h\,+\,5\)
You see, if you do the three steps
separately, it makes more sense ... and it's easier.
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** . IMPORTANT!
At this point, <u>two</u> things should have happened:
. . All of \(\displaystyle f(x)\) should have subtracted out.
. . Each remaining term must contain an \(\displaystyle h\).
If this is not true,
go back . . . you've made an error!
The most common errors:
. . Squaring incorrectly: \(\displaystyle (x + h)^2\) is NOT \(\displaystyle x^2 + h^2\) !!
. . Subtracting incorrectly: forgetting to "take the minus through".
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I have
always taught the Difference Quotient as a three-step recipe.
And I have
always pointed out the two Important warning signs.
From the number of related postings, I get the sinking feeling that
. . I'm the only teacher in the Alpha Quadrant that does this.