Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply.

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Step 3.4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.4.3.2

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

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

Step 3.4.4

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

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

Step 3.4.5

Simplify the expression.

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Step 3.4.5.1

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

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

Step 3.4.5.2

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

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

Step 3.4.6

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

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

Step 3.4.7

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

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

Step 3.4.8

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

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

Step 3.4.9

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

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

Step 3.5

Simplify.

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Reorder terms.

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

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

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

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Step 3.5.4.1.1

Apply the distributive property.

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

Step 3.5.4.1.2

Apply the distributive property.

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

Step 3.5.4.1.3

Apply the distributive property.

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

Step 3.5.4.2

Simplify and combine like terms.

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Step 3.5.4.2.1

Simplify each term.

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Step 3.5.4.2.1.1

Multiply $x$ by $1$ .

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

Step 3.5.4.2.1.2

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

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

Step 3.5.4.2.1.3

Cancel the common factor of $x$ .

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Step 3.5.4.2.1.3.1

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

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

Step 3.5.4.2.1.3.2

Cancel the common factor.

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

Step 3.5.4.2.1.3.3

Rewrite the expression.

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

Step 3.5.4.2.1.4

Combine.

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

Step 3.5.4.2.1.5

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

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Step 3.5.4.2.1.5.1

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

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Step 3.5.4.2.1.5.1.1

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

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

Step 3.5.4.2.1.5.1.2

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

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

Step 3.5.4.2.1.5.2

Add $2$ and $1$ .

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

Step 3.5.4.2.1.6

Multiply $1$ by $1$ .

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

Step 3.5.4.2.2

Add $\frac{1}{x}$ and $\frac{1}{x}$ .

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

Step 3.5.4.3

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

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

Step 3.5.4.4

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

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Step 3.5.4.4.1

Apply the distributive property.

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

Step 3.5.4.4.2

Apply the distributive property.

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

Step 3.5.4.4.3

Apply the distributive property.

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

Step 3.5.4.5

Simplify and combine like terms.

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Step 3.5.4.5.1

Simplify each term.

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Step 3.5.4.5.1.1

Multiply $x$ by $1$ .

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

Step 3.5.4.5.1.2

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

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

Step 3.5.4.5.1.3

Cancel the common factor of $x$ .

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Step 3.5.4.5.1.3.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

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

Step 3.5.4.5.1.3.2

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

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

Step 3.5.4.5.1.3.3

Cancel the common factor.

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

Step 3.5.4.5.1.3.4

Rewrite the expression.

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

Step 3.5.4.5.1.4

Move the negative in front of the fraction.

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

Step 3.5.4.5.1.5

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

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Step 3.5.4.5.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.4.5.1.5.2

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

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

Step 3.5.4.5.1.5.3

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

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

Step 3.5.4.5.1.5.4

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

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Step 3.5.4.5.1.5.4.1

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

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Step 3.5.4.5.1.5.4.1.1

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

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

Step 3.5.4.5.1.5.4.1.2

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

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

Step 3.5.4.5.1.5.4.2

Add $2$ and $1$ .

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

Step 3.5.4.5.2

Subtract $\frac{1}{x}$ from $- \frac{1}{x}$ .

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

Step 3.5.4.6

Simplify each term.

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Step 3.5.4.6.1

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

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

Step 3.5.4.6.2

Move the negative in front of the fraction.

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

Step 3.5.5

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

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Step 3.5.5.1

Subtract $\frac{2}{x}$ from $\frac{2}{x}$ .

$x + \frac{1}{x^{3}} + x + 0 + \frac{1}{x^{3}}$

Step 3.5.5.2

Add $x + \frac{1}{x^{3}} + x$ and $0$ .

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

Step 3.5.6

Combine the numerators over the common denominator.

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

Step 3.5.7

Add $1$ and $1$ .

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

Step 3.5.8

Add $x$ and $x$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 2 x + \frac{2}{x^{3}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 2 x + \frac{2}{x^{3}}$