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

find the difference quotient \\(\\frac{f(x+h)-f(x)}{h}\\), where \\(h \\neq 0\\), for the function \\(f(x) = 3x^2 - 12x\\). simplify your answer as much as possible.

find the difference quotient \\(\\frac{f(x+h)-f(x)}{h}\\), where \\(h \\neq 0\\), for the function \\(f(x) = 3x^2 - 12x\\). simplify your answer as much as possible.

Answer

Explanation:

Step1: Find ( f(x + h) )

Substitute ( x + h ) into ( f(x) = 3x^2 - 12x ): ( f(x + h) = 3(x + h)^2 - 12(x + h) ) Expand ( (x + h)^2 ) and distribute: ( f(x + h) = 3(x^2 + 2xh + h^2) - 12x - 12h ) ( f(x + h) = 3x^2 + 6xh + 3h^2 - 12x - 12h )

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

Subtract ( f(x) = 3x^2 - 12x ) from ( f(x + h) ): ( f(x + h) - f(x) = (3x^2 + 6xh + 3h^2 - 12x - 12h) - (3x^2 - 12x) ) Simplify by canceling ( 3x^2 ) and ( -12x ): ( f(x + h) - f(x) = 6xh + 3h^2 - 12h )

Step3: Divide by ( h ) ( ( h \neq 0 ) )

Divide ( f(x + h) - f(x) ) by ( h ): ( \frac{f(x + h) - f(x)}{h} = \frac{6xh + 3h^2 - 12h}{h} ) Factor out ( h ) in the numerator: ( \frac{f(x + h) - f(x)}{h} = \frac{h(6x + 3h - 12)}{h} ) Cancel ( h ) (since ( h \neq 0 )): ( \frac{f(x + h) - f(x)}{h} = 6x + 3h - 12 )

Answer:

( 6x + 3h - 12 )