find the difference quotient of f, that is, find \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \\neq…

find the difference quotient of f, that is, find \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \\neq 0 \\), for the following function. \\( f ( x ) = 8 x + 5 \\) \\( \\frac { f ( x + h ) - f ( x ) } { h } = \\) (simplify your answer.)

find the difference quotient of f, that is, find \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \\neq 0 \\), for the following function. \\( f ( x ) = 8 x + 5 \\) \\( \\frac { f ( x + h ) - f ( x ) } { h } = \\) (simplify your answer.)

Answer

Explanation:

Step1: Find ( f(x + h) )

Given ( f(x)=8x + 5 ), substitute ( x) with (x + h) in the function. ( f(x + h)=8(x + h)+5=8x+8h + 5 )

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

The difference - quotient formula is (\frac{f(x + h)-f(x)}{h}). Substitute ( f(x + h)=8x + 8h+5) and ( f(x)=8x + 5) into it: [ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{(8x + 8h+5)-(8x + 5)}{h}\ &=\frac{8x+8h + 5-8x - 5}{h} \end{align*} ]

Step3: Simplify the numerator and then the fraction

Simplify the numerator: (8x+8h + 5-8x - 5 = 8h) So, (\frac{8h}{h}), and since (h\neq0), we can cancel out the (h) terms.

Answer:

(8)