find the difference quotient $\\frac{f(x + h)-f(x)}{h}$, where $h\\neq0$, for the function…

find the difference quotient $\\frac{f(x + h)-f(x)}{h}$, where $h\\neq0$, for the function below.\n$f(x)=5x^{2}+7$\nsimplify your answer as much as possible.\n$\\frac{f(x + h)-f(x)}{h}=\\square$

find the difference quotient $\\frac{f(x + h)-f(x)}{h}$, where $h\\neq0$, for the function below.\n$f(x)=5x^{2}+7$\nsimplify your answer as much as possible.\n$\\frac{f(x + h)-f(x)}{h}=\\square$

Answer

Explanation:

Step1: Find ( f(x + h) )

Substitute ( x+h ) into ( f(x)=5x^{2}+7 ). ( f(x + h)=5(x + h)^{2}+7 ). Expand ( (x + h)^{2} ) using the formula ( (a + b)^{2}=a^{2}+2ab + b^{2} ), so ( f(x + h)=5(x^{2}+2xh+h^{2})+7=5x^{2}+10xh + 5h^{2}+7 ).

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

( f(x + h)-f(x)=(5x^{2}+10xh + 5h^{2}+7)-(5x^{2}+7) ). Remove the parentheses: ( f(x + h)-f(x)=5x^{2}+10xh + 5h^{2}+7 - 5x^{2}-7 ). Simplify by combining like - terms: ( f(x + h)-f(x)=10xh+5h^{2} ).

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

Substitute ( f(x + h)-f(x)=10xh + 5h^{2} ) into ( \frac{f(x + h)-f(x)}{h} ). ( \frac{f(x + h)-f(x)}{h}=\frac{10xh+5h^{2}}{h} ). Factor out ( h ) from the numerator: ( \frac{f(x + h)-f(x)}{h}=\frac{h(10x + 5h)}{h} ). Since ( h\neq0 ), cancel out the ( h ) terms.

Answer:

(10x + 5h)