Calculus Examples

$f (x) = \frac{6 x^{3} - 7 x^{2}}{\sqrt[3]{x^{4}}}$

Step 1

Rewrite $\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{\sqrt[3]{x^{4}}}]$ as $\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{x \sqrt[3]{x}}]$ .

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Step 1.1

Factor out $x^{3}$ .

$\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{\sqrt[3]{x^{3} x}}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{x \sqrt[3]{x}}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{x \cdot x^{\frac{1}{3}}}]$

Step 3

Multiply $x$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 3.1

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

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Step 3.1.1

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

$\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{x^{1} x^{\frac{1}{3}}}]$

Step 3.1.2

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

$\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{x^{1 + \frac{1}{3}}}]$

Step 3.2

Write $1$ as a fraction with a common denominator.

$\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{x^{\frac{3}{3} + \frac{1}{3}}}]$

Step 3.3

Combine the numerators over the common denominator.

$\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{x^{\frac{3 + 1}{3}}}]$

Step 3.4

Add $3$ and $1$ .

$\frac{d}{d x} [\frac{6 x^{3} - 7 x^{2}}{x^{\frac{4}{3}}}]$

Step 4

Factor $x^{2}$ out of $6 x^{3} - 7 x^{2}$ .

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Step 4.1

Factor $x^{2}$ out of $6 x^{3}$ .

$\frac{d}{d x} [\frac{x^{2} (6 x) - 7 x^{2}}{x^{\frac{4}{3}}}]$

Step 4.2

Factor $x^{2}$ out of $- 7 x^{2}$ .

$\frac{d}{d x} [\frac{x^{2} (6 x) + x^{2} \cdot - 7}{x^{\frac{4}{3}}}]$

Step 4.3

Factor $x^{2}$ out of $x^{2} (6 x) + x^{2} \cdot - 7$ .

$\frac{d}{d x} [\frac{x^{2} (6 x - 7)}{x^{\frac{4}{3}}}]$

Step 5

Move $x^{\frac{4}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{d}{d x} [x^{2} (6 x - 7) x^{- \frac{4}{3}}]$

Step 6

Multiply $x^{2}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 6.1

Move $x^{- \frac{4}{3}}$ .

$\frac{d}{d x} [x^{- \frac{4}{3}} x^{2} (6 x - 7)]$

Step 6.2

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

$\frac{d}{d x} [x^{- \frac{4}{3} + 2} (6 x - 7)]$

Step 6.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{d}{d x} [x^{- \frac{4}{3} + 2 \cdot \frac{3}{3}} (6 x - 7)]$

Step 6.4

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

$\frac{d}{d x} [x^{- \frac{4}{3} + \frac{2 \cdot 3}{3}} (6 x - 7)]$

Step 6.5

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{- 4 + 2 \cdot 3}{3}} (6 x - 7)]$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Multiply $2$ by $3$ .

$\frac{d}{d x} [x^{\frac{- 4 + 6}{3}} (6 x - 7)]$

Step 6.6.2

Add $- 4$ and $6$ .

$\frac{d}{d x} [x^{\frac{2}{3}} (6 x - 7)]$

Step 7

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 $f (x) = x^{\frac{2}{3}}$ and $g (x) = 6 x - 7$ .

$x^{\frac{2}{3}} \frac{d}{d x} [6 x - 7] + (6 x - 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 8

Differentiate.

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Step 8.1

By the Sum Rule, the derivative of $6 x - 7$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]$ .

$x^{\frac{2}{3}} (\frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]) + (6 x - 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 8.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$x^{\frac{2}{3}} (6 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]) + (6 x - 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 8.3

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

$x^{\frac{2}{3}} (6 \cdot 1 + \frac{d}{d x} [- 7]) + (6 x - 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 8.4

Multiply $6$ by $1$ .

$x^{\frac{2}{3}} (6 + \frac{d}{d x} [- 7]) + (6 x - 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 8.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$x^{\frac{2}{3}} (6 + 0) + (6 x - 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 8.6

Simplify the expression.

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Step 8.6.1

Add $6$ and $0$ .

$x^{\frac{2}{3}} \cdot 6 + (6 x - 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 8.6.2

Move $6$ to the left of $x^{\frac{2}{3}}$ .

$6 \cdot x^{\frac{2}{3}} + (6 x - 7) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 8.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$6 x^{\frac{2}{3}} + (6 x - 7) (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 x^{\frac{2}{3}} + (6 x - 7) (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 10

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

$6 x^{\frac{2}{3}} + (6 x - 7) (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 11

Combine the numerators over the common denominator.

$6 x^{\frac{2}{3}} + (6 x - 7) (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})$

Step 12

Simplify the numerator.

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Step 12.1

Multiply $- 1$ by $3$ .

$6 x^{\frac{2}{3}} + (6 x - 7) (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 12.2

Subtract $3$ from $2$ .

$6 x^{\frac{2}{3}} + (6 x - 7) (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 13

Move the negative in front of the fraction.

$6 x^{\frac{2}{3}} + (6 x - 7) (\frac{2}{3} x^{- \frac{1}{3}})$

Step 14

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

$6 x^{\frac{2}{3}} + (6 x - 7) \frac{2 x^{- \frac{1}{3}}}{3}$

Step 15

Move $x^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 x^{\frac{2}{3}} + (6 x - 7) \frac{2}{3 x^{\frac{1}{3}}}$

Step 16

Simplify.

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Step 16.1

Apply the distributive property.

$6 x^{\frac{2}{3}} + 6 x \frac{2}{3 x^{\frac{1}{3}}} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2

Combine terms.

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Step 16.2.1

Combine $6$ and $\frac{2}{3 x^{\frac{1}{3}}}$ .

$6 x^{\frac{2}{3}} + x \frac{6 \cdot 2}{3 x^{\frac{1}{3}}} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.2

Multiply $6$ by $2$ .

$6 x^{\frac{2}{3}} + x \frac{12}{3 x^{\frac{1}{3}}} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.3

Combine $x$ and $\frac{12}{3 x^{\frac{1}{3}}}$ .

$6 x^{\frac{2}{3}} + \frac{x \cdot 12}{3 x^{\frac{1}{3}}} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.4

Move $12$ to the left of $x$ .

$6 x^{\frac{2}{3}} + \frac{12 \cdot x}{3 x^{\frac{1}{3}}} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.5

Move $x^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$6 x^{\frac{2}{3}} + \frac{12 x \cdot x^{- \frac{1}{3}}}{3} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.6

Multiply $x$ by $x^{- \frac{1}{3}}$ by adding the exponents.

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Step 16.2.6.1

Move $x^{- \frac{1}{3}}$ .

$6 x^{\frac{2}{3}} + \frac{12 (x^{- \frac{1}{3}} x)}{3} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.6.2

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

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Step 16.2.6.2.1

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

$6 x^{\frac{2}{3}} + \frac{12 (x^{- \frac{1}{3}} x^{1})}{3} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.6.2.2

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

$6 x^{\frac{2}{3}} + \frac{12 x^{- \frac{1}{3} + 1}}{3} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.6.3

Write $1$ as a fraction with a common denominator.

$6 x^{\frac{2}{3}} + \frac{12 x^{- \frac{1}{3} + \frac{3}{3}}}{3} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.6.4

Combine the numerators over the common denominator.

$6 x^{\frac{2}{3}} + \frac{12 x^{\frac{- 1 + 3}{3}}}{3} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.6.5

Add $- 1$ and $3$ .

$6 x^{\frac{2}{3}} + \frac{12 x^{\frac{2}{3}}}{3} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.7

Factor $3$ out of $12 x^{\frac{2}{3}}$ .

$6 x^{\frac{2}{3}} + \frac{3 (4 x^{\frac{2}{3}})}{3} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.8

Cancel the common factors.

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Step 16.2.8.1

Factor $3$ out of $3$ .

$6 x^{\frac{2}{3}} + \frac{3 (4 x^{\frac{2}{3}})}{3 (1)} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.8.2

Cancel the common factor.

$6 x^{\frac{2}{3}} + \frac{3 (4 x^{\frac{2}{3}})}{3 \cdot 1} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.8.3

Rewrite the expression.

$6 x^{\frac{2}{3}} + \frac{4 x^{\frac{2}{3}}}{1} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.8.4

Divide $4 x^{\frac{2}{3}}$ by $1$ .

$6 x^{\frac{2}{3}} + 4 x^{\frac{2}{3}} - 7 \frac{2}{3 x^{\frac{1}{3}}}$

Step 16.2.9

Combine $- 7$ and $\frac{2}{3 x^{\frac{1}{3}}}$ .

$6 x^{\frac{2}{3}} + 4 x^{\frac{2}{3}} + \frac{- 7 \cdot 2}{3 x^{\frac{1}{3}}}$

Step 16.2.10

Multiply $- 7$ by $2$ .

$6 x^{\frac{2}{3}} + 4 x^{\frac{2}{3}} + \frac{- 14}{3 x^{\frac{1}{3}}}$

Step 16.2.11

Move the negative in front of the fraction.

$6 x^{\frac{2}{3}} + 4 x^{\frac{2}{3}} - \frac{14}{3 x^{\frac{1}{3}}}$

Step 16.2.12

Add $6 x^{\frac{2}{3}}$ and $4 x^{\frac{2}{3}}$ .

$10 x^{\frac{2}{3}} - \frac{14}{3 x^{\frac{1}{3}}}$