Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $x - \frac{1}{x}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $x - \frac{1}{x}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

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 $f (x) = - 1$ and $g (x) = \frac{1}{x}$ .

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

Step 4

Differentiate.

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Step 4.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 4.2

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

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

Step 4.3

Multiply.

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2

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

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

Step 4.4

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

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

Step 4.5

Simplify the expression.

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Step 4.5.1

Multiply $\frac{1}{x}$ by $0$ .

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

Step 4.5.2

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Combine terms.

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Step 5.3.1

Multiply $- 1$ by $2$ .

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

Step 5.3.2

Combine $- 2$ and $\frac{1}{x}$ .

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 5.4

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

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

Step 5.5

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

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Step 5.5.1

Apply the distributive property.

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

Step 5.5.2

Apply the distributive property.

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

Step 5.5.3

Apply the distributive property.

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

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

Multiply $2 x$ by $1$ .

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

Step 5.6.1.2

Multiply $- \frac{2}{x}$ by $1$ .

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

Step 5.6.1.3

Rewrite using the commutative property of multiplication.

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

Step 5.6.1.4

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

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

Step 5.6.1.5

Cancel the common factor of $x$ .

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Step 5.6.1.5.1

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

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

Step 5.6.1.5.2

Cancel the common factor.

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

Step 5.6.1.5.3

Rewrite the expression.

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

Step 5.6.1.6

Rewrite using the commutative property of multiplication.

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

Step 5.6.1.7

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

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Step 5.6.1.7.1

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

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

Step 5.6.1.7.2

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

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Step 5.6.1.7.2.1

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

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Step 5.6.1.7.2.1.1

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

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

Step 5.6.1.7.2.1.2

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

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

Step 5.6.1.7.2.2

Add $1$ and $2$ .

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

Step 5.6.2

Add $- \frac{2}{x}$ and $\frac{2}{x}$ .

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

Step 5.6.3

Add $2 x$ and $0$ .

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