simplify the difference quotient f(x + h) - f(x)/h for the given function. f(x)=6x^2 - 5x + 5 f(x + h)…

Explanation:

Step1: Find f(x + h)

Substitute (x+h) into (f(x)=6x^{2}-5x + 5). [ \begin{align*} f(x + h)&=6(x + h)^{2}-5(x + h)+5\ &=6(x^{2}+2xh+h^{2})-5x-5h + 5\ &=6x^{2}+12xh+6h^{2}-5x-5h + 5 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(6x^{2}+12xh+6h^{2}-5x-5h + 5)-(6x^{2}-5x + 5)\ &=6x^{2}+12xh+6h^{2}-5x-5h + 5-6x^{2}+5x - 5\ &=12xh+6h^{2}-5h \end{align*} ]

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

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{12xh+6h^{2}-5h}{h}\ &=\frac{h(12x + 6h-5)}{h}\ &=12x+6h - 5 \end{align*} ]

Answer:

(12x+6h - 5)