Question

the expression (\frac{f(x + h)-f(x)}{h}) for (h
eq0) is called the difference quotient. find and simplify the difference quotient for the following function. (f(x)=3x^{2}+10x + 10). the difference quotient is (simplify your answer.)

Explanation:

Step1: Find f(x + h)

Substitute x + h into f(x):
\[

$$\begin{align*} f(x + h)&=3(x + h)^2+10(x + h)+10\\ &=3(x^{2}+2xh+h^{2})+10x + 10h+10\\ &=3x^{2}+6xh+3h^{2}+10x + 10h+10 \end{align*}$$

\]

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

\[

$$\begin{align*} f(x + h)-f(x)&=(3x^{2}+6xh+3h^{2}+10x + 10h+10)-(3x^{2}+10x + 10)\\ &=3x^{2}+6xh+3h^{2}+10x + 10h+10 - 3x^{2}-10x - 10\\ &=6xh+3h^{2}+10h \end{align*}$$

\]

Step3: Calculate the difference - quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{6xh+3h^{2}+10h}{h}\\ &=\frac{h(6x + 3h+10)}{h}\\ &=6x + 3h+10 \end{align*}$$

\]

Answer:

$6x + 3h+10$