# For the function f(x)= x^2+8x, how do you find and simplify f(x+h)-f(x)?

Nov 12, 2016

${h}^{2} + 2 x h - 8 x$

#### Explanation:

$f \left(x\right) = {x}^{2} + 8 x$ for variable unknown $x$

If the variable unknown is $x + h$ then the function $f$ will become:
$f \left(x + h\right) = {\left(x + h\right)}^{2} + 8 \left(x + h\right)$

Thus, $f \left(x + h\right) - f \left(x\right)$ is:

$\left[{\left(x + h\right)}^{2} + 8 \left(x + h\right)\right] - \left[{x}^{2} + 8 x\right]$
$\left[\left(x + h\right) \left(x + h\right) + 8 x + 8 h\right] - {x}^{2} - 8 x$
$\left[{x}^{2} + 2 x h + {h}^{2}\right] - {x}^{2} - 8 x$

Cancel out any corresponding terms:
$\setminus \cancel{{x}^{2}} + 2 x h + {h}^{2} \setminus \cancel{- {x}^{2}} - 8 x \setminus \rightarrow \setminus \textcolor{red}{{h}^{2} + 2 x h - 8 x}$