Calculus Examples

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

Step 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 $f (x) = x^{2} + 1$ and $g (x) = x^{2} - 1$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

Move $2$ to the left of $x^{2} - 1$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $2 x$ and $0$ .

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

Step 2.8.2

Multiply $2$ by $- 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Combine the opposite terms in $2 x^{2} x + 2 \cdot - 1 x - 2 x^{2} x - 2 \cdot 1 x$ .

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Step 3.5.1.1

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

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

Step 3.5.1.2

Add $2 \cdot - 1 x$ and $0$ .

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

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

Multiply $2$ by $- 1$ .

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

Step 3.5.2.2

Multiply $- 2$ by $1$ .

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

Step 3.5.3

Subtract $2 x$ from $- 2 x$ .

$\frac{- 4 x}{{(x^{2} - 1)}^{2}}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{4 x}{{(x^{2} - 1)}^{2}}$

Step 3.7

Simplify the denominator.

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Step 3.7.1

Rewrite $1$ as $1^{2}$ .

$- \frac{4 x}{{(x^{2} - 1^{2})}^{2}}$

Step 3.7.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$- \frac{4 x}{{((x + 1) (x - 1))}^{2}}$

Step 3.7.3

Apply the product rule to $(x + 1) (x - 1)$ .

$- \frac{4 x}{{(x + 1)}^{2} {(x - 1)}^{2}}$