You have the following function:
[tex]f(x)=x^2+8[/tex]
In order to calculate
[tex]\frac{f(x+h)-f(x)}{h}[/tex]
Consider that f(x + h) is given by:
[tex]f(x+h)=(x+h)^2+8=x^2+2xh+h^2[/tex]
Then, by replacing f(x) and f(x+h) into the difference quotient and by simplifying, you get:
[tex]\begin{gathered} \frac{f(x+h)-f(x)}{h}=\frac{x^2+2xh+h^2+8-x^2-8}{h} \\ \frac{f(x+h)-f(x)}{h}=\frac{2xh+h^2}{h}=\frac{h(2x+h)}{h}=2x+h \end{gathered}[/tex]
Hence, the result to the quotient difference is 2x + h
