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.\n\\( f ( x ) = - x ^ { 2 } + 4 x + 5 \\)\n\\( \\frac { f ( x + h ) - f ( x ) } { h } = \\square \\)

find and simplify the difference quotient \\( \\frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \\neq 0 \\) for the given function.\n\\( f ( x ) = - x ^ { 2 } + 4 x + 5 \\)\n\\( \\frac { f ( x + h ) - f ( x ) } { h } = \\square \\)

Answer

Explanation:

Step1: Find ( f(x + h) )

Substitute ( x+h ) into ( f(x)=-x^{2}+4x + 5 ). [ \begin{align*} f(x + h)&=-(x + h)^{2}+4(x + h)+5\ &=-(x^{2}+2xh+h^{2})+4x + 4h+5\ &=-x^{2}-2xh - h^{2}+4x + 4h+5 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(-x^{2}-2xh - h^{2}+4x + 4h+5)-(-x^{2}+4x + 5)\ &=-x^{2}-2xh - h^{2}+4x + 4h+5 + x^{2}-4x - 5\ &=-2xh - h^{2}+4h \end{align*} ]

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

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-2xh - h^{2}+4h}{h}\ &=\frac{h(-2x - h + 4)}{h}\ &=-2x - h+4 \end{align*} ]

Answer:

(-2x - h + 4)