consider the function below.\n$f(x)=5x^{3}+6x$\nfind the difference quotient below (where $h\neq0$) and…

Explanation:

Step1: Find (f(x + h))

Substitute (x+h) into (f(x)): [ \begin{align*} f(x + h)&=5(x + h)^{3}+6(x + h)\ &=5(x^{3}+3x^{2}h + 3xh^{2}+h^{3})+6x+6h\ &=5x^{3}+15x^{2}h+15xh^{2}+5h^{3}+6x + 6h \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(5x^{3}+15x^{2}h+15xh^{2}+5h^{3}+6x + 6h)-(5x^{3}+6x)\ &=5x^{3}+15x^{2}h+15xh^{2}+5h^{3}+6x + 6h - 5x^{3}-6x\ &=15x^{2}h+15xh^{2}+5h^{3}+6h \end{align*} ]

Step3: Find the difference - quotient (\frac{f(x + h)-f(x)}{h})

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{15x^{2}h+15xh^{2}+5h^{3}+6h}{h}\ &=\frac{h(15x^{2}+15xh + 5h^{2}+6)}{h}\ &=15x^{2}+15xh+5h^{2}+6 \end{align*} ]

Answer:

(15x^{2}+15xh + 5h^{2}+6)