Question

find and simplify the difference quotient $\frac{f(x + h)-f(x)}{h}$, $h
eq0$ for the given function. $f(x)=x^{2}+5x + 6$
a. 1
b. $2x+h + 6$
c. $\frac{2x^{2}+2x+2xh+h^{2}+h + 12}{h}$
d. $2x+h + 5$

Explanation:

Step1: Find f(x + h)

$f(x + h)=(x + h)^2+5(x + h)+6=x^{2}+2xh+h^{2}+5x + 5h+6$

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

$f(x + h)-f(x)=(x^{2}+2xh+h^{2}+5x + 5h+6)-(x^{2}+5x + 6)=2xh+h^{2}+5h$

Step3: Find the difference quotient

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

Answer:

D. 2x + h + 5