find the difference quotient $\frac{f(x + h)-f(x)}{h}$, where $h\neq0$, for the function below.\n\n$f(x)=5x^{…

find the difference quotient $\frac{f(x + h)-f(x)}{h}$, where $h\neq0$, for the function below.\n\n$f(x)=5x^{2}+4x$\n\nsimplify your answer as much as possible.\n\n$\frac{f(x + h)-f(x)}{h}=$

find the difference quotient $\frac{f(x + h)-f(x)}{h}$, where $h\neq0$, for the function below.\n\n$f(x)=5x^{2}+4x$\n\nsimplify your answer as much as possible.\n\n$\frac{f(x + h)-f(x)}{h}=$

Answer

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=5(x + h)^2+4(x + h)\ &=5(x^{2}+2xh+h^{2})+4x + 4h\ &=5x^{2}+10xh+5h^{2}+4x + 4h \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(5x^{2}+10xh+5h^{2}+4x + 4h)-(5x^{2}+4x)\ &=5x^{2}+10xh+5h^{2}+4x + 4h - 5x^{2}-4x\ &=10xh+5h^{2}+4h \end{align*} ]

Step3: Find the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{10xh+5h^{2}+4h}{h}\ &=\frac{h(10x + 5h+4)}{h}\ &=10x+5h + 4 \end{align*} ]

Answer:

$10x + 5h+4$