Calculus Examples

$f (x) = \frac{4 - 3 x - x^{2}}{x^{2} - 1}$

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 $f (x) = 4 - 3 x - x^{2}$ and $g (x) = x^{2} - 1$ .

$\frac{(x^{2} - 1) \frac{d}{d x} [4 - 3 x - x^{2}] - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 - 3 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- x^{2}]$ .

$\frac{(x^{2} - 1) (\frac{d}{d x} [4] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- x^{2}]) - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 1) (0 + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- x^{2}]) - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2.3

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$\frac{(x^{2} - 1) (\frac{d}{d x} [- 3 x] + \frac{d}{d x} [- x^{2}]) - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$\frac{(x^{2} - 1) (- 3 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}]) - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2.5

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

$\frac{(x^{2} - 1) (- 3 \cdot 1 + \frac{d}{d x} [- x^{2}]) - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2.6

Multiply $- 3$ by $1$ .

$\frac{(x^{2} - 1) (- 3 + \frac{d}{d x} [- x^{2}]) - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$\frac{(x^{2} - 1) (- 3 - \frac{d}{d x} [x^{2}]) - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2.8

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

$\frac{(x^{2} - 1) (- 3 - (2 x)) - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2.9

Multiply $2$ by $- 1$ .

$\frac{(x^{2} - 1) (- 3 - 2 x) - (4 - 3 x - x^{2}) \frac{d}{d x} [x^{2} - 1]}{{(x^{2} - 1)}^{2}}$

Step 2.10

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

$\frac{(x^{2} - 1) (- 3 - 2 x) - (4 - 3 x - x^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

Step 2.11

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

$\frac{(x^{2} - 1) (- 3 - 2 x) - (4 - 3 x - x^{2}) (2 x + \frac{d}{d x} [- 1])}{{(x^{2} - 1)}^{2}}$

Step 2.12

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

$\frac{(x^{2} - 1) (- 3 - 2 x) - (4 - 3 x - x^{2}) (2 x + 0)}{{(x^{2} - 1)}^{2}}$

Step 2.13

Simplify the expression.

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Step 2.13.1

Add $2 x$ and $0$ .

$\frac{(x^{2} - 1) (- 3 - 2 x) - (4 - 3 x - x^{2}) (2 x)}{{(x^{2} - 1)}^{2}}$

Step 2.13.2

Multiply $2$ by $- 1$ .

$\frac{(x^{2} - 1) (- 3 - 2 x) - 2 (4 - 3 x - x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(x^{2} - 1) (- 3 - 2 x) + (- 2 \cdot 4 - 2 (- 3 x) - 2 (- x^{2})) x}{{(x^{2} - 1)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{(x^{2} - 1) (- 3 - 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Expand $(x^{2} - 1) (- 3 - 2 x)$ using the FOIL Method.

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Step 3.3.1.1.1

Apply the distributive property.

$\frac{x^{2} (- 3 - 2 x) - 1 (- 3 - 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.1.2

Apply the distributive property.

$\frac{x^{2} \cdot - 3 + x^{2} (- 2 x) - 1 (- 3 - 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.1.3

Apply the distributive property.

$\frac{x^{2} \cdot - 3 + x^{2} (- 2 x) - 1 \cdot - 3 - 1 (- 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.2

Simplify each term.

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Step 3.3.1.2.1

Move $- 3$ to the left of $x^{2}$ .

$\frac{- 3 \cdot x^{2} + x^{2} (- 2 x) - 1 \cdot - 3 - 1 (- 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.2.2

Rewrite using the commutative property of multiplication.

$\frac{- 3 x^{2} - 2 x^{2} x - 1 \cdot - 3 - 1 (- 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.2.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.3.1.2.3.1

Move $x$ .

$\frac{- 3 x^{2} - 2 (x \cdot x^{2}) - 1 \cdot - 3 - 1 (- 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.2.3.2

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

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Step 3.3.1.2.3.2.1

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

$\frac{- 3 x^{2} - 2 (x^{1} x^{2}) - 1 \cdot - 3 - 1 (- 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.2.3.2.2

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

$\frac{- 3 x^{2} - 2 x^{1 + 2} - 1 \cdot - 3 - 1 (- 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.2.3.3

Add $1$ and $2$ .

$\frac{- 3 x^{2} - 2 x^{3} - 1 \cdot - 3 - 1 (- 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.2.4

Multiply $- 1$ by $- 3$ .

$\frac{- 3 x^{2} - 2 x^{3} + 3 - 1 (- 2 x) - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.2.5

Multiply $- 2$ by $- 1$ .

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 2 \cdot 4 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.3

Multiply $- 2$ by $4$ .

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x - 2 (- 3 x) x - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.4.1

Move $x$ .

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x - 2 (- 3 (x \cdot x)) - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.4.2

Multiply $x$ by $x$ .

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x - 2 (- 3 x^{2}) - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.5

Multiply $- 3$ by $- 2$ .

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x + 6 x^{2} - 2 (- x^{2}) x}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.3.1.6.1

Move $x$ .

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x + 6 x^{2} - 2 (- (x \cdot x^{2}))}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.6.2

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

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Step 3.3.1.6.2.1

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

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x + 6 x^{2} - 2 (- (x^{1} x^{2}))}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.6.2.2

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

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x + 6 x^{2} - 2 (- x^{1 + 2})}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.6.3

Add $1$ and $2$ .

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x + 6 x^{2} - 2 (- x^{3})}{{(x^{2} - 1)}^{2}}$

Step 3.3.1.7

Multiply $- 1$ by $- 2$ .

$\frac{- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x + 6 x^{2} + 2 x^{3}}{{(x^{2} - 1)}^{2}}$

Step 3.3.2

Combine the opposite terms in $- 3 x^{2} - 2 x^{3} + 3 + 2 x - 8 x + 6 x^{2} + 2 x^{3}$ .

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Step 3.3.2.1

Add $- 2 x^{3}$ and $2 x^{3}$ .

$\frac{- 3 x^{2} + 3 + 2 x - 8 x + 6 x^{2} + 0}{{(x^{2} - 1)}^{2}}$

Step 3.3.2.2

Add $- 3 x^{2} + 3 + 2 x - 8 x + 6 x^{2}$ and $0$ .

$\frac{- 3 x^{2} + 3 + 2 x - 8 x + 6 x^{2}}{{(x^{2} - 1)}^{2}}$

Step 3.3.3

Add $- 3 x^{2}$ and $6 x^{2}$ .

$\frac{3 x^{2} + 3 + 2 x - 8 x}{{(x^{2} - 1)}^{2}}$

Step 3.3.4

Subtract $8 x$ from $2 x$ .

$\frac{3 x^{2} + 3 - 6 x}{{(x^{2} - 1)}^{2}}$

Step 3.4

Reorder terms.

$\frac{3 x^{2} - 6 x + 3}{{(x^{2} - 1)}^{2}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $3$ out of $3 x^{2} - 6 x + 3$ .

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Step 3.5.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) - 6 x + 3}{{(x^{2} - 1)}^{2}}$

Step 3.5.1.2

Factor $3$ out of $- 6 x$ .

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

Step 3.5.1.3

Factor $3$ out of $3$ .

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

Step 3.5.1.4

Factor $3$ out of $3 (x^{2}) + 3 (- 2 x)$ .

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

Step 3.5.1.5

Factor $3$ out of $3 (x^{2} - 2 x) + 3 (1)$ .

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

Step 3.5.2

Factor using the perfect square rule.

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Step 3.5.2.1

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

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

Step 3.5.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 3.5.2.3

Rewrite the polynomial.

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

Step 3.5.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 3.6

Simplify the denominator.

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Step 3.6.1

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

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

Step 3.6.2

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 $b = 1$ .

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

Step 3.6.3

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

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

Step 3.7

Cancel the common factor of ${(x - 1)}^{2}$ .

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Step 3.7.1

Cancel the common factor.

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

Step 3.7.2

Rewrite the expression.

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