Question

for the function $f(x)=x^{2}+8$, construct and simplify the difference quotient $\frac{f(x + h)-f(x)}{h}$. the difference quotient is

Explanation:

Step1: Find \( f(x + h) \)

Substitute \( x+h \) into \( f(x)=x^{2}+8 \).
\( f(x + h)=(x + h)^{2}+8=x^{2}+2xh+h^{2}+8 \)

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

\( f(x + h)-f(x)=(x^{2}+2xh+h^{2}+8)-(x^{2}+8) \)
\( =x^{2}+2xh+h^{2}+8 - x^{2}-8=2xh + h^{2} \)

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

\( \frac{f(x + h)-f(x)}{h}=\frac{2xh + h^{2}}{h}=\frac{h(2x + h)}{h}=2x+h \)

Answer:

\( 2x + h \)