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

find and simplify the difference quotient \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \\neq 0 \\) for the given function. \\( f ( x ) = 3 x ^ { 2 } \\) \\( \\frac { f ( x + h ) - f ( x ) } { h } = \\square \\) (simplify your answer.)

find and simplify the difference quotient \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \\neq 0 \\) for the given function. \\( f ( x ) = 3 x ^ { 2 } \\) \\( \\frac { f ( x + h ) - f ( x ) } { h } = \\square \\) (simplify your answer.)

Answer

Explanation:

Step1: Find ( f(x + h) )

Given ( f(x)=3x^{2} ), substitute ( x) with ( x + h): ( f(x + h)=3(x + h)^{2}=3(x^{2}+2xh+h^{2})=3x^{2}+6xh + 3h^{2} )

Step2: Substitute ( f(x + h) ) and ( f(x) ) into the difference quotient formula

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

Step3: Simplify the numerator and then divide by ( h )

The ( 3x^{2}) terms in the numerator cancel out: ( \frac{6xh+3h^{2}}{h} ). Factor out ( h ) from the numerator: ( \frac{h(6x + 3h)}{h} ). Since ( h\neq0 ), we can cancel out the ( h ) terms: ( 6x+3h )

Answer:

(6x + 3h)