find the difference quotient $\frac{f(x + h)-f(x)}{h}$ for the function and simplify it. f(x)=9x^2 + 3x the…

Explanation:

Step1: Find f(x + h)

Substitute (x+h) into (f(x)): [ \begin{align*} f(x + h)&=9(x + h)^2+3(x + h)\ &=9(x^{2}+2xh+h^{2})+3x + 3h\ &=9x^{2}+18xh+9h^{2}+3x + 3h \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(9x^{2}+18xh+9h^{2}+3x + 3h)-(9x^{2}+3x)\ &=9x^{2}+18xh+9h^{2}+3x + 3h - 9x^{2}-3x\ &=18xh+9h^{2}+3h \end{align*} ]

Step3: Find the difference quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{18xh+9h^{2}+3h}{h}\ &=\frac{h(18x + 9h+3)}{h}\ &=18x+9h + 3 \end{align*} ]

Answer:

(18x + 9h+3)