Question

find the difference quotient of f(x)=x^2 - 1; that is find $\frac{f(x + h)-f(x)}{h}$, h≠0. be sure to simplify. the difference quotient is □.

Explanation:

Step1: Find f(x + h)

Given \(f(x)=x^{2}-1\), then \(f(x + h)=(x + h)^{2}-1=x^{2}+2xh+h^{2}-1\).

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

\[

$$\begin{align*} f(x + h)-f(x)&=(x^{2}+2xh + h^{2}-1)-(x^{2}-1)\\ &=x^{2}+2xh+h^{2}-1 - x^{2}+1\\ &=2xh+h^{2} \end{align*}$$

\]

Step3: Compute the difference - quotient

\(\frac{f(x + h)-f(x)}{h}=\frac{2xh + h^{2}}{h}\). Since \(h
eq0\), we can factor out \(h\) from the numerator and cancel it with the denominator: \(\frac{h(2x + h)}{h}=2x+h\).

Answer:

\(2x + h\)