Find f(x+h) - f(x) for f(x) = x (squared) +5x - 1
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Hello, wayneo1688!
I assume you're new to the Difference Quotient.
Don't let the Difference Quotient scare you.Find $\displaystyle \frac{f(x+h) - f(x)}{h}$ . for . $\displaystyle f(x) \:= \:x^2 +5x - 1$
. . Break it down into its three basic steps.
(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 $\displaystyle h$ . . . Factor and cancel (carefully!)
Here we go! . . . We have: .$\displaystyle f(x)\:=\:x^2 +5x - 1$
$\displaystyle (1)\;\;f(x+h) \;=\;(x+h)^2 + 5(x+h) - 1 \;=\;x^2 + 2xh + h^2 + 5x + 5h - 1$
$\displaystyle (2)\;\;f(x+h) - f(x) \;=\;(x^2 + 2xh + h^2 _ 5x + 5h - 1) - (x^2 + 5x-1) \;=\;2xh + h^2 + 5h$
$\displaystyle (3)\;\;\frac{f(x+h)-f(x)}{h}\;=\;\frac{2xh + h^2 + 5h}{h} \;=\;\frac{\not{h\;}(2x + h + 5)}{\not{h\:}} \;=\;{\color{blue}2x + h + 5}$