find the difference quotient $\\frac{f(x + h)-f(x)}{h}$, where $h\\neq0$, for the function…

find the difference quotient $\\frac{f(x + h)-f(x)}{h}$, where $h\\neq0$, for the function below.\n$f(x)=5x^{2}-5x + 2$\nsimplify your answer as much as possible.\n$\\frac{f(x + h)-f(x)}{h}=\\square$

find the difference quotient $\\frac{f(x + h)-f(x)}{h}$, where $h\\neq0$, for the function below.\n$f(x)=5x^{2}-5x + 2$\nsimplify your answer as much as possible.\n$\\frac{f(x + h)-f(x)}{h}=\\square$

Answer

Explanation:

Step1: Find ( f(x + h) )

Substitute ( x+h ) into ( f(x)=5x^{2}-5x + 2 ): [ \begin{align*} f(x + h)&=5(x + h)^{2}-5(x + h)+2\ &=5(x^{2}+2xh+h^{2})-5x-5h + 2\ &=5x^{2}+10xh+5h^{2}-5x-5h + 2 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(5x^{2}+10xh+5h^{2}-5x-5h + 2)-(5x^{2}-5x + 2)\ &=5x^{2}+10xh+5h^{2}-5x-5h + 2-5x^{2}+5x - 2\ &=10xh+5h^{2}-5h \end{align*} ]

Step3: Compute ( \frac{f(x + h)-f(x)}{h} )

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{10xh+5h^{2}-5h}{h}\ &=\frac{h(10x + 5h-5)}{h}\ &=10x+5h - 5 \end{align*} ]

Answer:

(10x + 5h-5)