Question

Use the definition of the derivative to find $f$for each function $f$in Exercises 39-54.

$f\left(x\right)=\frac{x^2-1}{x^2-x-2}$

Short answer

The derivative is$f^{\text{'}}\left(x\right)=\frac{2x\left(x^2-x-2\right)-(2x-1)\left(x^2-1\right)}{{\left(x^2-x-2\right)}^2}$

Step-by-step solution

Given information

Given function$f\left(x\right)=\frac{x^2-1}{x^2-x-2}$

Use the chain rule and calculate

Calculating, we get

$$\begin{gathered} f\left(x\right)=\frac{x^2-1}{x^2-x-2} \\ f\text{'}\left(x\right)=\frac{(x^2-x-2)\left(2x\right)-(x^2-1)(2x-1)}{{(x^2-x-2)}^2} \\ f\text{'}\left(x\right)=\frac{2x(x^2-x-2)-(2x-1)(x^2-1)}{{(x^2-x-2)}^2} \end{gathered}$$