find the difference quotient \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), where \\( h \\neq 0 \\), for…

find the difference quotient \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), where \\( h \\neq 0 \\), for the function below.\n\\( f ( x ) = 3 x ^ { 2 } - 5 x + 2 \\)\nsimplify your answer as much as possible.\n\\( \\frac { f ( x + h ) - f ( x ) } { h } = \\)

find the difference quotient \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), where \\( h \\neq 0 \\), for the function below.\n\\( f ( x ) = 3 x ^ { 2 } - 5 x + 2 \\)\nsimplify your answer as much as possible.\n\\( \\frac { f ( x + h ) - f ( x ) } { h } = \\)

Answer

Explanation:

Step1: Find ( f(x + h) )

Substitute ( x+h ) into ( f(x)=3x^{2}-5x + 2 ). [ \begin{align*} f(x + h)&=3(x + h)^{2}-5(x + h)+2\ &=3(x^{2}+2xh+h^{2})-5x-5h + 2\ &=3x^{2}+6xh+3h^{2}-5x-5h + 2 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(3x^{2}+6xh+3h^{2}-5x-5h + 2)-(3x^{2}-5x + 2)\ &=3x^{2}+6xh+3h^{2}-5x-5h + 2-3x^{2}+5x - 2\ &=6xh+3h^{2}-5h \end{align*} ]

Step3: Find the difference quotient (\frac{f(x + h)-f(x)}{h})

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{6xh+3h^{2}-5h}{h}\ &=\frac{h(6x + 3h-5)}{h}\ &=6x+3h - 5 \end{align*} ]

Answer:

(6x + 3h-5)