Q. 54 - Calculus

Question

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

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

Step-by-Step Solution

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 1: Given information

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

Step 2: Use the chain rule and calculate

Calculating, we get

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