Calculus Examples

$(\frac{3}{x} + x) (\frac{3}{x} - x)$

Step 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 $f (x) = \frac{3}{x} + x$ and $g (x) = \frac{3}{x} - x$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $- 1$ by $3$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Multiply $- 1$ by $1$ .

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

Step 2.9

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

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

Step 2.10

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

Multiply $- 1$ by $3$ .

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

Step 2.14

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Combine terms.

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Step 3.3.1

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

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

Step 3.3.2

Move the negative in front of the fraction.

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

Step 3.3.3

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

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

Step 3.3.4

Move the negative in front of the fraction.

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

Step 3.4

Reorder terms.

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

Step 3.5

Simplify each term.

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Step 3.5.1

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

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Step 3.5.1.1

Apply the distributive property.

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

Step 3.5.1.2

Apply the distributive property.

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

Step 3.5.1.3

Apply the distributive property.

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

Step 3.5.2

Simplify and combine like terms.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

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

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Step 3.5.2.1.1.1

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

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

Step 3.5.2.1.1.2

Multiply $3$ by $3$ .

$- \frac{9}{x \cdot x^{2}} - \frac{3}{x^{2}} x - 1 \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.1.3

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

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Step 3.5.2.1.1.3.1

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

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Step 3.5.2.1.1.3.1.1

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

$- \frac{9}{x^{1} x^{2}} - \frac{3}{x^{2}} x - 1 \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.1.3.1.2

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

$- \frac{9}{x^{1 + 2}} - \frac{3}{x^{2}} x - 1 \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.1.3.2

Add $1$ and $2$ .

$- \frac{9}{x^{3}} - \frac{3}{x^{2}} x - 1 \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.2

Cancel the common factor of $x$ .

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Step 3.5.2.1.2.1

Move the leading negative in $- \frac{3}{x^{2}}$ into the numerator.

$- \frac{9}{x^{3}} + \frac{- 3}{x^{2}} x - 1 \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.2.2

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

$- \frac{9}{x^{3}} + \frac{- 3}{x \cdot x} x - 1 \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.2.3

Cancel the common factor.

$- \frac{9}{x^{3}} + \frac{- 3}{x \cdot x} x - 1 \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.2.4

Rewrite the expression.

$- \frac{9}{x^{3}} + \frac{- 3}{x} - 1 \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.3

Move the negative in front of the fraction.

$- \frac{9}{x^{3}} - \frac{3}{x} - 1 \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.4

Rewrite $- 1 \frac{3}{x}$ as $- \frac{3}{x}$ .

$- \frac{9}{x^{3}} - \frac{3}{x} - \frac{3}{x} - 1 x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.1.5

Rewrite $- 1 x$ as $- x$ .

$- \frac{9}{x^{3}} - \frac{3}{x} - \frac{3}{x} - x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.2.2

Subtract $\frac{3}{x}$ from $- \frac{3}{x}$ .

$- \frac{9}{x^{3}} - 2 \frac{3}{x} - x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.3

Simplify each term.

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Step 3.5.3.1

Multiply $- 2 \frac{3}{x}$ .

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Step 3.5.3.1.1

Combine $- 2$ and $\frac{3}{x}$ .

$- \frac{9}{x^{3}} + \frac{- 2 \cdot 3}{x} - x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.3.1.2

Multiply $- 2$ by $3$ .

$- \frac{9}{x^{3}} + \frac{- 6}{x} - x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.3.2

Move the negative in front of the fraction.

$- \frac{9}{x^{3}} - \frac{6}{x} - x + (- \frac{3}{x^{2}} + 1) (\frac{3}{x} - x)$

Step 3.5.4

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

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Step 3.5.4.1

Apply the distributive property.

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{3}{x^{2}} (\frac{3}{x} - x) + 1 (\frac{3}{x} - x)$

Step 3.5.4.2

Apply the distributive property.

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{3}{x^{2}} \cdot \frac{3}{x} - \frac{3}{x^{2}} (- x) + 1 (\frac{3}{x} - x)$

Step 3.5.4.3

Apply the distributive property.

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{3}{x^{2}} \cdot \frac{3}{x} - \frac{3}{x^{2}} (- x) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5

Simplify and combine like terms.

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Step 3.5.5.1

Simplify each term.

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Step 3.5.5.1.1

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

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Step 3.5.5.1.1.1

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

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{3 \cdot 3}{x \cdot x^{2}} - \frac{3}{x^{2}} (- x) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.1.2

Multiply $3$ by $3$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x \cdot x^{2}} - \frac{3}{x^{2}} (- x) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.1.3

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

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Step 3.5.5.1.1.3.1

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

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Step 3.5.5.1.1.3.1.1

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

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{1} x^{2}} - \frac{3}{x^{2}} (- x) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.1.3.1.2

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

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{1 + 2}} - \frac{3}{x^{2}} (- x) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.1.3.2

Add $1$ and $2$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} - \frac{3}{x^{2}} (- x) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.2

Cancel the common factor of $x$ .

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Step 3.5.5.1.2.1

Move the leading negative in $- \frac{3}{x^{2}}$ into the numerator.

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{- 3}{x^{2}} (- x) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.2.2

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

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{- 3}{x \cdot x} (- x) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.2.3

Factor $x$ out of $- x$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{- 3}{x \cdot x} (x \cdot - 1) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.2.4

Cancel the common factor.

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{- 3}{x \cdot x} (x \cdot - 1) + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.2.5

Rewrite the expression.

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{- 3}{x} \cdot - 1 + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.3

Combine $\frac{- 3}{x}$ and $- 1$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{- 3 \cdot - 1}{x} + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.4

Multiply $- 3$ by $- 1$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{3}{x} + 1 \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.5

Multiply $\frac{3}{x}$ by $1$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{3}{x} + \frac{3}{x} + 1 (- x)$

Step 3.5.5.1.6

Multiply $- x$ by $1$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{3}{x} + \frac{3}{x} - x$

Step 3.5.5.2

Add $\frac{3}{x}$ and $\frac{3}{x}$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + 2 \frac{3}{x} - x$

Step 3.5.6

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

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Step 3.5.6.1

Combine $2$ and $\frac{3}{x}$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{2 \cdot 3}{x} - x$

Step 3.5.6.2

Multiply $2$ by $3$ .

$- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{6}{x} - x$

Step 3.6

Combine the opposite terms in $- \frac{9}{x^{3}} - \frac{6}{x} - x - \frac{9}{x^{3}} + \frac{6}{x} - x$ .

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Step 3.6.1

Add $- \frac{6}{x}$ and $\frac{6}{x}$ .

$- \frac{9}{x^{3}} - x - \frac{9}{x^{3}} + 0 - x$

Step 3.6.2

Add $- \frac{9}{x^{3}} - x - \frac{9}{x^{3}}$ and $0$ .

$- \frac{9}{x^{3}} - x - \frac{9}{x^{3}} - x$

Step 3.7

Combine the numerators over the common denominator.

$- x - x + \frac{- 9 - 9}{x^{3}}$

Step 3.8

Subtract $9$ from $- 9$ .

$- x - x + \frac{- 18}{x^{3}}$

Step 3.9

Move the negative in front of the fraction.

$- x - x - \frac{18}{x^{3}}$

Step 3.10

Subtract $x$ from $- x$ .

$- 2 x - \frac{18}{x^{3}}$