Find the Derivative f(x)=x/(x^2-1)

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

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = x^{2} - 1$ .

$\frac{(x^{2} - 1) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Differentiate.

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Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{(x^{2} - 1) \cdot 1 - x \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Multiply $x^{2} - 1$ by $1$ .

$\frac{x^{2} - 1 - x \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$\frac{x^{2} - 1 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{x^{2} - 1 - x (2 x + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{x^{2} - 1 - x (2 x + 0)}{{(x^{2} - 1)}^{2}}$

Simplify the expression.

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Add $2 x$ and $0$ .

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

Multiply $2$ by $- 1$ .

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

Raise $x$ to the power of $1$ .

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

Raise $x$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Subtract $2 x^{2}$ from $x^{2}$ .

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

Simplify.

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Factor $- 1$ out of $- x^{2}$ .

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

Rewrite $- 1$ as $- 1 (1)$ .

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

Factor $- 1$ out of $- (x^{2}) - 1 (1)$ .

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

Rewrite $- (x^{2} + 1)$ as $- 1 (x^{2} + 1)$ .

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

Move the negative in front of the fraction.

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

Find the Derivative f(x)=x/(x^2-1)

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