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. be sure to simplify. \\( f ( x ) = x ^ { 2 } - 2 x + 6 \\) \\( \\frac { f ( x + h ) - f ( x ) } { h } = \\square \\) (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. be sure to simplify. \\( f ( x ) = x ^ { 2 } - 2 x + 6 \\) \\( \\frac { f ( x + h ) - f ( x ) } { h } = \\square \\) (simplify your answer.)

Answer

Explanation:

Step1: Find ( f(x + h) )

Substitute ( x + h ) into ( f(x) ): ( f(x + h)=(x + h)^2-2(x + h)+6=x^{2}+2xh+h^{2}-2x - 2h+6 )

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

( f(x + h)-f(x)=(x^{2}+2xh+h^{2}-2x - 2h+6)-(x^{2}-2x + 6) ) ( =x^{2}+2xh+h^{2}-2x - 2h+6 - x^{2}+2x - 6 ) ( =2xh+h^{2}-2h )

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

( \frac{f(x + h)-f(x)}{h}=\frac{2xh+h^{2}-2h}{h} ) Factor out ( h ) in the numerator: ( \frac{h(2x + h-2)}{h} ) Since ( h\neq0 ), cancel out ( h ): ( 2x+h - 2 )

Answer:

( 2x+h - 2 )