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

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

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

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

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

Answer:

(-2x - h + 5)