Calculus Examples

$\frac{x + 3}{x^{2} - 1}$

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 $f (x) = x + 3$ and $g (x) = x^{2} - 1$ .

$\frac{(x^{2} - 1) \frac{d}{d x} [x + 3] - (x + 3) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Differentiate.

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By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{(x^{2} - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

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

$\frac{(x^{2} - 1) (1 + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 1) (1 + 0) - (x + 3) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Simplify the expression.

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Add $1$ and $0$ to get $1$ .

$\frac{(x^{2} - 1) \cdot 1 - (x + 3) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

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

$\frac{x^{2} - 1 - (x + 3) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

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

$\frac{x^{2} - 1 - (x + 3) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

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

$\frac{x^{2} - 1 - (x + 3) (2 x + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

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

$\frac{x^{2} - 1 - (x + 3) (2 x + 0)}{{(x^{2} - 1)}^{2}}$

Simplify the expression.

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Add $2 x$ and $0$ to get $2 x$ .

$\frac{x^{2} - 1 - (x + 3) (2 x)}{{(x^{2} - 1)}^{2}}$

Multiply $2$ by $- 1$ to get $- 2$ .

$\frac{x^{2} - 1 - 2 (x + 3) x}{{(x^{2} - 1)}^{2}}$

Simplify.

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Apply the distributive property.

$\frac{x^{2} - 1 + (- 2 x - 2 \cdot 3) x}{{(x^{2} - 1)}^{2}}$

Apply the distributive property.

$\frac{x^{2} - 1 + (- 2 x x - 2 \cdot 3 x)}{{(x^{2} - 1)}^{2}}$

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

$\frac{x^{2} - 1 + (- 2 (x^{1} x) - 2 \cdot 3 x)}{{(x^{2} - 1)}^{2}}$

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

$\frac{x^{2} - 1 + (- 2 (x^{1} x^{1}) - 2 \cdot 3 x)}{{(x^{2} - 1)}^{2}}$

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

$\frac{x^{2} - 1 + (- 2 x^{1 + 1} - 2 \cdot 3 x)}{{(x^{2} - 1)}^{2}}$

Add $1$ and $1$ to get $2$ .

$\frac{x^{2} - 1 + (- 2 x^{2} - 2 \cdot 3 x)}{{(x^{2} - 1)}^{2}}$

Multiply $- 2$ by $3$ to get $- 6$ .

$\frac{x^{2} - 1 + (- 2 x^{2} - 6 x)}{{(x^{2} - 1)}^{2}}$

Remove parentheses.

$\frac{x^{2} - 1 - 2 x^{2} - 6 x}{{(x^{2} - 1)}^{2}}$

Subtract $2 x^{2}$ from $x^{2}$ to get $- x^{2}$ .

$\frac{- x^{2} - 1 - 6 x}{{(x^{2} - 1)}^{2}}$

Reorder terms.

$\frac{- x^{2} - 6 x - 1}{{(x^{2} - 1)}^{2}}$

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

$\frac{- (x^{2}) - 6 x - 1}{{(x^{2} - 1)}^{2}}$

Factor $- 1$ out of $- 6 x$ .

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Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- (x^{2}) - 1 (6) x - 1}{{(x^{2} - 1)}^{2}}$

Move $- 1$ .

$\frac{- (x^{2}) - ((6) x) - 1}{{(x^{2} - 1)}^{2}}$

Multiply $6$ by $x$ to get $6 x$ .

$\frac{- (x^{2}) - (6 x) - 1}{{(x^{2} - 1)}^{2}}$

Factor $- 1$ out of $- (x^{2}) - (6 x)$ .

$\frac{- (x^{2} + 6 x) - 1}{{(x^{2} - 1)}^{2}}$

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

$\frac{- (x^{2} + 6 x) - 1 (1)}{{(x^{2} - 1)}^{2}}$

Factor $- 1$ out of $- (x^{2} + 6 x) - 1 (1)$ .

$\frac{- (x^{2} + 6 x + 1)}{{(x^{2} - 1)}^{2}}$

Rewrite $- (x^{2} + 6 x + 1)$ as $- 1 (x^{2} + 6 x + 1)$ .

$\frac{- 1 (x^{2} + 6 x + 1)}{{(x^{2} - 1)}^{2}}$

Move the negative in front of the fraction.

$- \frac{x^{2} + 6 x + 1}{{(x^{2} - 1)}^{2}}$

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