Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $(x^{2} - 1) (x^{2} + x + 1)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $(x^{2} - 1) (x^{2} + x + 1)$ .

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $2 x + 1$ and $0$ .

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

Simplify the expression.

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Step 4.9.1

Add $2 x$ and $0$ .

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

Step 4.9.2

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

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

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.2

Apply the product rule to $(x^{2} - 1) (x^{2} + x + 1)$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Combine terms.

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

Add $1$ and $2$ .

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

Step 5.5.4

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

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

Step 5.5.5

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

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

Step 5.5.6

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

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

Step 5.5.7

Add $1$ and $1$ .

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

Step 5.5.8

Multiply $2$ by $1$ .

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

Step 5.6

Reorder the factors of $- \frac{1}{{(x^{2} - 1)}^{2} {(x^{2} + x + 1)}^{2}} ((x^{2} - 1) (2 x + 1) + 2 x^{3} + 2 x^{2} + 2 x)$ .

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

Step 5.7

Simplify each term.

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Step 5.7.1

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

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Step 5.7.1.1

Apply the distributive property.

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

Step 5.7.1.2

Apply the distributive property.

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

Step 5.7.1.3

Apply the distributive property.

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

Step 5.7.2

Simplify each term.

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Step 5.7.2.1

Rewrite using the commutative property of multiplication.

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

Step 5.7.2.2

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

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Step 5.7.2.2.1

Move $x$ .

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

Step 5.7.2.2.2

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

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Step 5.7.2.2.2.1

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

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

Step 5.7.2.2.2.2

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

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

Step 5.7.2.2.3

Add $1$ and $2$ .

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

Step 5.7.2.3

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

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

Step 5.7.2.4

Multiply $2$ by $- 1$ .

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

Step 5.7.2.5

Multiply $- 1$ by $1$ .

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

Step 5.8

Combine the opposite terms in $2 x^{3} + x^{2} - 2 x - 1 + 2 x^{3} + 2 x^{2} + 2 x$ .

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Step 5.8.1

Add $- 2 x$ and $2 x$ .

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

Step 5.8.2

Add $2 x^{3} + x^{2} - 1 + 2 x^{3} + 2 x^{2}$ and $0$ .

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

Apply the distributive property.

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

Step 5.12

Simplify.

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Step 5.12.1

Multiply $4$ by $- 1$ .

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

Step 5.12.2

Multiply $3$ by $- 1$ .

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

Step 5.12.3

Multiply $- 1$ by $- 1$ .

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

Step 5.13

Simplify the denominator.

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Step 5.13.1

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

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

Step 5.13.2

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 $b = 1$ .

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

Step 5.13.3

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

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

Step 5.14

Multiply $- 4 x^{3} - 3 x^{2} + 1$ by $\frac{1}{{(x + 1)}^{2} {(x - 1)}^{2} {(x^{2} + x + 1)}^{2}}$ .

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

Step 5.15

Factor $- 1$ out of $- 4 x^{3}$ .

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

Step 5.16

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

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

Step 5.17

Factor $- 1$ out of $- (4 x^{3}) - (3 x^{2})$ .

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

Step 5.18

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

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

Step 5.19

Factor $- 1$ out of $- (4 x^{3} + 3 x^{2}) - 1 (- 1)$ .

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

Step 5.20

Rewrite $- (4 x^{3} + 3 x^{2} - 1)$ as $- 1 (4 x^{3} + 3 x^{2} - 1)$ .

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

Step 5.21

Move the negative in front of the fraction.

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