Question

find the difference quotient $\frac{f(x + h)-f(x)}{h}$ for the function and simplify.
$f(x)=x^{2}+5x$
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Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$:
$f(x + h)=(x + h)^2+5(x + h)=x^{2}+2xh+h^{2}+5x + 5h$

Step2: Calculate $f(x + h)-f(x)$

$f(x + h)-f(x)=(x^{2}+2xh+h^{2}+5x + 5h)-(x^{2}+5x)$
$=x^{2}+2xh+h^{2}+5x + 5h - x^{2}-5x=2xh+h^{2}+5h$

Step3: Find the difference quotient

$\frac{f(x + h)-f(x)}{h}=\frac{2xh+h^{2}+5h}{h}$
Factor out $h$ from the numerator: $\frac{h(2x + h+5)}{h}$
Cancel out $h$ (assuming $h
eq0$): $2x+h + 5$

Answer:

$2x+h + 5$