Question

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

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

Short answer

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

Step-by-step solution

Given information

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

Use the rule to calculate the derivative

Calculating, we get

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