Calculus Examples

{(x + x^{- 1})}^{3}

${(x + x^{- 1})}^{3}$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = x^{3}

$f (x) = x^{3}$ and

g (x) = x + x^{- 1}

$g (x) = x + x^{- 1}$ .

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Step 1.1

To apply the Chain Rule, set

u

$u$ as

x + x^{- 1}

$x + x^{- 1}$ .

\frac{d}{d u} [u^{3}] \frac{d}{d x} [x + x^{- 1}]

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [x + x^{- 1}]$

Step 1.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = 3

$n = 3$ .

3 u^{2} \frac{d}{d x} [x + x^{- 1}]

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

Step 1.3

Replace all occurrences of

u

$u$ with

x + x^{- 1}

$x + x^{- 1}$ .

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

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

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

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of

x + x^{- 1}

$x + x^{- 1}$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [x^{- 1}]

$\frac{d}{d x} [x] + \frac{d}{d x} [x^{- 1}]$ .

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

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

Step 2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

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

Step 2.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = - 1

$n = - 1$ .

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

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

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

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

Step 3

Simplify.

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Step 3.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 3.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 3.3

Rewrite

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

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

(x + \frac{1}{x}) (x + \frac{1}{x})

$(x + \frac{1}{x}) (x + \frac{1}{x})$ .

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

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

Step 3.4

Expand

(x + \frac{1}{x}) (x + \frac{1}{x})

$(x + \frac{1}{x}) (x + \frac{1}{x})$ using the FOIL Method.

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Step 3.4.1

Apply the distributive property.

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

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

Step 3.4.2

Apply the distributive property.

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

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

Step 3.4.3

Apply the distributive property.

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

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

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

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

Step 3.5

Simplify and combine like terms.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply

x

$x$ by

x

$x$ .

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

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

Step 3.5.1.2

Cancel the common factor of

x

$x$ .

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Step 3.5.1.2.1

Cancel the common factor.

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

Step 3.5.1.2.2

Rewrite the expression.

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

Step 3.5.1.3

Cancel the common factor of

$x$ .

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Step 3.5.1.3.1

Cancel the common factor.

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

Step 3.5.1.3.2

Rewrite the expression.

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

Step 3.5.1.4

Multiply

$\frac{1}{x} \cdot \frac{1}{x}$ .

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Step 3.5.1.4.1

Multiply

$\frac{1}{x}$ by

$\frac{1}{x}$ .

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

Step 3.5.1.4.2

Raise

$x$ to the power of

$1$ .

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

Step 3.5.1.4.3

Raise

$x$ to the power of

$1$ .

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

Step 3.5.1.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.1.4.5

Add

$1$ and

$1$ .

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

Step 3.5.2

Add

$1$ and

$1$ .

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Simplify.

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Step 3.7.1

Multiply

$3$ by

$2$ .

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

Step 3.7.2

Combine

$3$ and

$\frac{1}{x^{2}}$ .

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

Step 3.8

Expand

$(3 x^{2} + 6 + \frac{3}{x^{2}}) (1 - \frac{1}{x^{2}})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.9

Simplify each term.

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Step 3.9.1

Multiply

$3$ by

$1$ .

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

Step 3.9.2

Cancel the common factor of

$x^{2}$ .

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Step 3.9.2.1

Move the leading negative in

$- \frac{1}{x^{2}}$ into the numerator.

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

Step 3.9.2.2

Factor

$x^{2}$ out of

$3 x^{2}$ .

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

Step 3.9.2.3

Cancel the common factor.

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

Step 3.9.2.4

Rewrite the expression.

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

Step 3.9.3

Multiply

$3$ by

$- 1$ .

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

Step 3.9.4

Multiply

$6$ by

$1$ .

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

Step 3.9.5

Multiply

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

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Step 3.9.5.1

Multiply

$- 1$ by

$6$ .

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

Step 3.9.5.2

Combine

$- 6$ and

$\frac{1}{x^{2}}$ .

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

Step 3.9.6

Move the negative in front of the fraction.

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

Step 3.9.7

Multiply

$\frac{3}{x^{2}}$ by

$1$ .

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

Step 3.9.8

Rewrite using the commutative property of multiplication.

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

Step 3.9.9

Multiply

$- \frac{3}{x^{2}} \cdot \frac{1}{x^{2}}$ .

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Step 3.9.9.1

Multiply

$\frac{1}{x^{2}}$ by

$\frac{3}{x^{2}}$ .

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

Step 3.9.9.2

Multiply

$x^{2}$ by

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

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Step 3.9.9.2.1

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.9.9.2.2

Add

$2$ and

$2$ .

$3 x^{2} - 3 + 6 - \frac{6}{x^{2}} + \frac{3}{x^{2}} - \frac{3}{x^{4}}$

Step 3.10

Combine the numerators over the common denominator.

$3 x^{2} - 3 + 6 + \frac{- 6 + 3}{x^{2}} + \frac{- 3}{x^{4}}$

Step 3.11

Add

$- 6$ and

$3$ .

$3 x^{2} - 3 + 6 + \frac{- 3}{x^{2}} + \frac{- 3}{x^{4}}$

Step 3.12

Simplify each term.

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Step 3.12.1

Move the negative in front of the fraction.

$3 x^{2} - 3 + 6 - \frac{3}{x^{2}} + \frac{- 3}{x^{4}}$

Step 3.12.2

Move the negative in front of the fraction.

$3 x^{2} - 3 + 6 - \frac{3}{x^{2}} - \frac{3}{x^{4}}$

Step 3.13

Add

$- 3$ and

$6$ .

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