the expression $\frac{f(x + h)-f(x)}{h}$ for $h\neq0$ is called the difference quotient. find and simplify…

the expression $\frac{f(x + h)-f(x)}{h}$ for $h\neq0$ is called the difference quotient. find and simplify the difference quotient for the following function. $f(x)=-9x^{2}+6x + 6$. the difference quotient is $square$. (simplify your answer.)

the expression $\frac{f(x + h)-f(x)}{h}$ for $h\neq0$ is called the difference quotient. find and simplify the difference quotient for the following function. $f(x)=-9x^{2}+6x + 6$. the difference quotient is $square$. (simplify your answer.)

Answer

Explanation:

Step1: Find f(x + h)

Substitute (x+h) into (f(x)=-9x^{2}+6x + 6). [ \begin{align*} f(x + h)&=-9(x + h)^{2}+6(x + h)+6\ &=-9(x^{2}+2xh+h^{2})+6x+6h + 6\ &=-9x^{2}-18xh-9h^{2}+6x+6h + 6 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(-9x^{2}-18xh-9h^{2}+6x+6h + 6)-(-9x^{2}+6x + 6)\ &=-9x^{2}-18xh-9h^{2}+6x+6h + 6 + 9x^{2}-6x - 6\ &=-18xh-9h^{2}+6h \end{align*} ]

Step3: Find the difference quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-18xh-9h^{2}+6h}{h}\ &=\frac{h(-18x-9h + 6)}{h}\ &=-18x-9h + 6 \end{align*} ]

Answer:

(-18x-9h + 6)