23. find the difference quotient $\frac{f(x + h)-f(x)}{h}$ for $f(x)=x^{2}-4$

Explanation:

Step1: Find (f(x + h))

Given (f(x)=x^{2}-4), then (f(x + h)=(x + h)^{2}-4=x^{2}+2xh+h^{2}-4)

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

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

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

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

Answer:

(2x + h)