# Step-by-step Solution

## Find the derivative using the quotient rule $\frac{d}{dx}\left(\frac{x}{x^2-1}\right)$

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### Videos

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

## Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\frac{x}{x^2-1}\right)$
1

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\left(x^2-1\right)\frac{d}{dx}\left(x\right)-x\frac{d}{dx}\left(x^2-1\right)}{\left(x^2-1\right)^2}$
2

The derivative of the linear function is equal to $1$

$\frac{x^2-1-x\frac{d}{dx}\left(x^2-1\right)}{\left(x^2-1\right)^2}$

$\frac{-1-x^2}{\left(x^2-1\right)^2}$
$\frac{d}{dx}\left(\frac{x}{x^2-1}\right)$

Quotient rule

~ 0.83 seconds

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