Calculus Examples

$y = (5 x - 3) (4 x - \frac{3}{x})$

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 $f (x) = 5 x - 3$ and $g (x) = 4 x - \frac{3}{x}$ .

$(5 x - 3) \frac{d}{d x} [4 x - \frac{3}{x}] + (4 x - \frac{3}{x}) \frac{d}{d x} [5 x - 3]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x - \frac{3}{x}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- \frac{3}{x}]$ .

$(5 x - 3) (\frac{d}{d x} [4 x] + \frac{d}{d x} [- \frac{3}{x}]) + (4 x - \frac{3}{x}) \frac{d}{d x} [5 x - 3]$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$(5 x - 3) (4 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{3}{x}]) + (4 x - \frac{3}{x}) \frac{d}{d x} [5 x - 3]$

Step 2.3

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

$(5 x - 3) (4 \cdot 1 + \frac{d}{d x} [- \frac{3}{x}]) + (4 x - \frac{3}{x}) \frac{d}{d x} [5 x - 3]$

Step 2.4

Multiply $4$ by $1$ .

$(5 x - 3) (4 + \frac{d}{d x} [- \frac{3}{x}]) + (4 x - \frac{3}{x}) \frac{d}{d x} [5 x - 3]$

Step 2.5

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

$(5 x - 3) (4 - 3 \frac{d}{d x} [\frac{1}{x}]) + (4 x - \frac{3}{x}) \frac{d}{d x} [5 x - 3]$

Step 2.6

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

$(5 x - 3) (4 - 3 \frac{d}{d x} [x^{- 1}]) + (4 x - \frac{3}{x}) \frac{d}{d x} [5 x - 3]$

Step 2.7

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

$(5 x - 3) (4 - 3 (- x^{- 2})) + (4 x - \frac{3}{x}) \frac{d}{d x} [5 x - 3]$

Step 2.8

Multiply $- 1$ by $- 3$ .

$(5 x - 3) (4 + 3 x^{- 2}) + (4 x - \frac{3}{x}) \frac{d}{d x} [5 x - 3]$

Step 2.9

By the Sum Rule, the derivative of $5 x - 3$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 3]$ .

$(5 x - 3) (4 + 3 x^{- 2}) + (4 x - \frac{3}{x}) (\frac{d}{d x} [5 x] + \frac{d}{d x} [- 3])$

Step 2.10

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$(5 x - 3) (4 + 3 x^{- 2}) + (4 x - \frac{3}{x}) (5 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 2.11

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

$(5 x - 3) (4 + 3 x^{- 2}) + (4 x - \frac{3}{x}) (5 \cdot 1 + \frac{d}{d x} [- 3])$

Step 2.12

Multiply $5$ by $1$ .

$(5 x - 3) (4 + 3 x^{- 2}) + (4 x - \frac{3}{x}) (5 + \frac{d}{d x} [- 3])$

Step 2.13

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(5 x - 3) (4 + 3 x^{- 2}) + (4 x - \frac{3}{x}) (5 + 0)$

Step 2.14

Simplify the expression.

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Step 2.14.1

Add $5$ and $0$ .

$(5 x - 3) (4 + 3 x^{- 2}) + (4 x - \frac{3}{x}) \cdot 5$

Step 2.14.2

Move $5$ to the left of $4 x - \frac{3}{x}$ .

$(5 x - 3) (4 + 3 x^{- 2}) + 5 \cdot (4 x - \frac{3}{x})$

Step 3

Simplify.

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Step 3.1

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

$(5 x - 3) (4 + 3 \frac{1}{x^{2}}) + 5 (4 x - \frac{3}{x})$

Step 3.2

Apply the distributive property.

$(5 x - 3) (4 + 3 \frac{1}{x^{2}}) + 5 (4 x) + 5 (- \frac{3}{x})$

Step 3.3

Combine terms.

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Step 3.3.1

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

$(5 x - 3) (4 + \frac{3}{x^{2}}) + 5 (4 x) + 5 (- \frac{3}{x})$

Step 3.3.2

Multiply $4$ by $5$ .

$(5 x - 3) (4 + \frac{3}{x^{2}}) + 20 x + 5 (- \frac{3}{x})$

Step 3.3.3

Multiply $- 1$ by $5$ .

$(5 x - 3) (4 + \frac{3}{x^{2}}) + 20 x - 5 \frac{3}{x}$

Step 3.3.4

Combine $- 5$ and $\frac{3}{x}$ .

$(5 x - 3) (4 + \frac{3}{x^{2}}) + 20 x + \frac{- 5 \cdot 3}{x}$

Step 3.3.5

Multiply $- 5$ by $3$ .

$(5 x - 3) (4 + \frac{3}{x^{2}}) + 20 x + \frac{- 15}{x}$

Step 3.3.6

Move the negative in front of the fraction.

$(5 x - 3) (4 + \frac{3}{x^{2}}) + 20 x - \frac{15}{x}$

Step 3.3.7

To write $(5 x - 3) (4 + \frac{3}{x^{2}})$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$20 x + \frac{(5 x - 3) (4 + \frac{3}{x^{2}}) x}{x} - \frac{15}{x}$

Step 3.3.8

Combine the numerators over the common denominator.

$20 x + \frac{(5 x - 3) (4 + \frac{3}{x^{2}}) x - 15}{x}$

Step 3.3.9

To write $20 x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{20 x \cdot x}{x} + \frac{(5 x - 3) (4 + \frac{3}{x^{2}}) x - 15}{x}$

Step 3.3.10

Combine the numerators over the common denominator.

$\frac{20 x \cdot x + (5 x - 3) (4 + \frac{3}{x^{2}}) x - 15}{x}$

Step 3.3.11

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

$\frac{20 (x^{1} x) + (5 x - 3) (4 + \frac{3}{x^{2}}) x - 15}{x}$

Step 3.3.12

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

$\frac{20 (x^{1} x^{1}) + (5 x - 3) (4 + \frac{3}{x^{2}}) x - 15}{x}$

Step 3.3.13

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

$\frac{20 x^{1 + 1} + (5 x - 3) (4 + \frac{3}{x^{2}}) x - 15}{x}$

Step 3.3.14

Add $1$ and $1$ .

$\frac{20 x^{2} + (5 x - 3) (4 + \frac{3}{x^{2}}) x - 15}{x}$

Step 3.4

Reorder terms.

$\frac{20 x^{2} + x (4 + \frac{3}{x^{2}}) (5 x - 3) - 15}{x}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Apply the distributive property.

$\frac{20 x^{2} + (x \cdot 4 + x \frac{3}{x^{2}}) (5 x - 3) - 15}{x}$

Step 3.5.2

Move $4$ to the left of $x$ .

$\frac{20 x^{2} + (4 \cdot x + x \frac{3}{x^{2}}) (5 x - 3) - 15}{x}$

Step 3.5.3

Cancel the common factor of $x$ .

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Step 3.5.3.1

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

$\frac{20 x^{2} + (4 \cdot x + x \frac{3}{x \cdot x}) (5 x - 3) - 15}{x}$

Step 3.5.3.2

Cancel the common factor.

$\frac{20 x^{2} + (4 \cdot x + x \frac{3}{x \cdot x}) (5 x - 3) - 15}{x}$

Step 3.5.3.3

Rewrite the expression.

$\frac{20 x^{2} + (4 \cdot x + \frac{3}{x}) (5 x - 3) - 15}{x}$

Step 3.5.4

Expand $(4 x + \frac{3}{x}) (5 x - 3)$ using the FOIL Method.

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Step 3.5.4.1

Apply the distributive property.

$\frac{20 x^{2} + 4 x (5 x - 3) + \frac{3}{x} (5 x - 3) - 15}{x}$

Step 3.5.4.2

Apply the distributive property.

$\frac{20 x^{2} + 4 x (5 x) + 4 x \cdot - 3 + \frac{3}{x} (5 x - 3) - 15}{x}$

Step 3.5.4.3

Apply the distributive property.

$\frac{20 x^{2} + 4 x (5 x) + 4 x \cdot - 3 + \frac{3}{x} (5 x) + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5

Simplify each term.

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Step 3.5.5.1

Rewrite using the commutative property of multiplication.

$\frac{20 x^{2} + 4 \cdot 5 x \cdot x + 4 x \cdot - 3 + \frac{3}{x} (5 x) + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.5.5.2.1

Move $x$ .

$\frac{20 x^{2} + 4 \cdot 5 (x \cdot x) + 4 x \cdot - 3 + \frac{3}{x} (5 x) + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.2.2

Multiply $x$ by $x$ .

$\frac{20 x^{2} + 4 \cdot 5 x^{2} + 4 x \cdot - 3 + \frac{3}{x} (5 x) + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.3

Multiply $4$ by $5$ .

$\frac{20 x^{2} + 20 x^{2} + 4 x \cdot - 3 + \frac{3}{x} (5 x) + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.4

Multiply $- 3$ by $4$ .

$\frac{20 x^{2} + 20 x^{2} - 12 x + \frac{3}{x} (5 x) + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.5

Rewrite using the commutative property of multiplication.

$\frac{20 x^{2} + 20 x^{2} - 12 x + 5 \frac{3}{x} x + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.6

Multiply $5 \frac{3}{x}$ .

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Step 3.5.5.6.1

Combine $5$ and $\frac{3}{x}$ .

$\frac{20 x^{2} + 20 x^{2} - 12 x + \frac{5 \cdot 3}{x} x + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.6.2

Multiply $5$ by $3$ .

$\frac{20 x^{2} + 20 x^{2} - 12 x + \frac{15}{x} x + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.7

Cancel the common factor of $x$ .

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Step 3.5.5.7.1

Cancel the common factor.

$\frac{20 x^{2} + 20 x^{2} - 12 x + \frac{15}{x} x + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.7.2

Rewrite the expression.

$\frac{20 x^{2} + 20 x^{2} - 12 x + 15 + \frac{3}{x} \cdot - 3 - 15}{x}$

Step 3.5.5.8

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

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Step 3.5.5.8.1

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

$\frac{20 x^{2} + 20 x^{2} - 12 x + 15 + \frac{3 \cdot - 3}{x} - 15}{x}$

Step 3.5.5.8.2

Multiply $3$ by $- 3$ .

$\frac{20 x^{2} + 20 x^{2} - 12 x + 15 + \frac{- 9}{x} - 15}{x}$

Step 3.5.5.9

Move the negative in front of the fraction.

$\frac{20 x^{2} + 20 x^{2} - 12 x + 15 - \frac{9}{x} - 15}{x}$

Step 3.5.6

Add $20 x^{2}$ and $20 x^{2}$ .

$\frac{40 x^{2} - 12 x + 15 - \frac{9}{x} - 15}{x}$

Step 3.5.7

Subtract $15$ from $15$ .

$\frac{40 x^{2} - 12 x - \frac{9}{x} + 0}{x}$

Step 3.5.8

Add $40 x^{2} - 12 x - \frac{9}{x}$ and $0$ .

$\frac{40 x^{2} - 12 x - \frac{9}{x}}{x}$

Step 3.5.9

To write $40 x^{2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{- 12 x + \frac{40 x^{2} x}{x} - \frac{9}{x}}{x}$

Step 3.5.10

Combine the numerators over the common denominator.

$\frac{- 12 x + \frac{40 x^{2} x - 9}{x}}{x}$

Step 3.5.11

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

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Step 3.5.11.1

Move $x$ .

$\frac{- 12 x + \frac{40 (x \cdot x^{2}) - 9}{x}}{x}$

Step 3.5.11.2

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

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Step 3.5.11.2.1

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

$\frac{- 12 x + \frac{40 (x^{1} x^{2}) - 9}{x}}{x}$

Step 3.5.11.2.2

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

$\frac{- 12 x + \frac{40 x^{1 + 2} - 9}{x}}{x}$

Step 3.5.11.3

Add $1$ and $2$ .

$\frac{- 12 x + \frac{40 x^{3} - 9}{x}}{x}$

Step 3.5.12

To write $- 12 x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{- 12 x \cdot \frac{x}{x} + \frac{40 x^{3} - 9}{x}}{x}$

Step 3.5.13

Combine $- 12 x$ and $\frac{x}{x}$ .

$\frac{\frac{- 12 x \cdot x}{x} + \frac{40 x^{3} - 9}{x}}{x}$

Step 3.5.14

Combine the numerators over the common denominator.

$\frac{\frac{- 12 x \cdot x + 40 x^{3} - 9}{x}}{x}$

Step 3.5.15

Simplify the numerator.

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Step 3.5.15.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.5.15.1.1

Move $x$ .

$\frac{\frac{- 12 (x \cdot x) + 40 x^{3} - 9}{x}}{x}$

Step 3.5.15.1.2

Multiply $x$ by $x$ .

$\frac{\frac{- 12 x^{2} + 40 x^{3} - 9}{x}}{x}$

Step 3.5.15.2

Reorder terms.

$\frac{\frac{40 x^{3} - 12 x^{2} - 9}{x}}{x}$

Step 3.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{40 x^{3} - 12 x^{2} - 9}{x} \cdot \frac{1}{x}$

Step 3.7

Multiply $\frac{40 x^{3} - 12 x^{2} - 9}{x} \cdot \frac{1}{x}$ .

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Step 3.7.1

Multiply $\frac{40 x^{3} - 12 x^{2} - 9}{x}$ by $\frac{1}{x}$ .

$\frac{40 x^{3} - 12 x^{2} - 9}{x \cdot x}$

Step 3.7.2

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

$\frac{40 x^{3} - 12 x^{2} - 9}{x^{1} x}$

Step 3.7.3

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

$\frac{40 x^{3} - 12 x^{2} - 9}{x^{1} x^{1}}$

Step 3.7.4

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

$\frac{40 x^{3} - 12 x^{2} - 9}{x^{1 + 1}}$

Step 3.7.5

Add $1$ and $1$ .

$\frac{40 x^{3} - 12 x^{2} - 9}{x^{2}}$