Calculus Examples

$y = \frac{v^{3} - 2 v \sqrt{v}}{v}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d v} [\frac{f (v)}{g (v)}]$ is $\frac{g (v) \frac{d}{d v} [f (v)] - f (v) \frac{d}{d v} [g (v)]}{{g (v)}^{2}}$ where $f (v) = v^{3} - 2 v \sqrt{v}$ and $g (v) = v$ .

$\frac{v \frac{d}{d v} [v^{3} - 2 v \sqrt{v}] - (v^{3} - 2 v \sqrt{v}) \frac{d}{d v} [v]}{v^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $v^{3} - 2 v \sqrt{v}$ with respect to $v$ is $\frac{d}{d v} [v^{3}] + \frac{d}{d v} [- 2 v \sqrt{v}]$ .

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

Step 2.2

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

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

Step 3

Evaluate $\frac{d}{d v} [- 2 v \sqrt{v}]$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{v}$ as $v^{\frac{1}{2}}$ .

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

Step 3.2

Multiply $v$ by $v^{\frac{1}{2}}$ by adding the exponents.

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Step 3.2.1

Move $v^{\frac{1}{2}}$ .

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Add $1$ and $2$ .

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

Step 3.3

Since $- 2$ is constant with respect to $v$ , the derivative of $- 2 v^{\frac{3}{2}}$ with respect to $v$ is $- 2 \frac{d}{d v} [v^{\frac{3}{2}}]$ .

$\frac{v (3 v^{2} - 2 \frac{d}{d v} [v^{\frac{3}{2}}]) - (v^{3} - 2 v \sqrt{v}) \frac{d}{d v} [v]}{v^{2}}$

Step 3.4

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

$\frac{v (3 v^{2} - 2 (\frac{3}{2} v^{\frac{3}{2} - 1})) - (v^{3} - 2 v \sqrt{v}) \frac{d}{d v} [v]}{v^{2}}$

Step 3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $2$ .

$\frac{v (3 v^{2} - 2 (\frac{3}{2} v^{\frac{3 - 2}{2}})) - (v^{3} - 2 v \sqrt{v}) \frac{d}{d v} [v]}{v^{2}}$

Step 3.8.2

Subtract $2$ from $3$ .

$\frac{v (3 v^{2} - 2 (\frac{3}{2} v^{\frac{1}{2}})) - (v^{3} - 2 v \sqrt{v}) \frac{d}{d v} [v]}{v^{2}}$

Step 3.9

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

$\frac{v (3 v^{2} - 2 \frac{3 v^{\frac{1}{2}}}{2}) - (v^{3} - 2 v \sqrt{v}) \frac{d}{d v} [v]}{v^{2}}$

Step 3.10

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

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

Step 3.11

Multiply $3$ by $- 2$ .

$\frac{v (3 v^{2} + \frac{- 6 v^{\frac{1}{2}}}{2}) - (v^{3} - 2 v \sqrt{v}) \frac{d}{d v} [v]}{v^{2}}$

Step 3.12

Factor $2$ out of $- 6 v^{\frac{1}{2}}$ .

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

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $2$ out of $2$ .

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

Step 3.13.2

Cancel the common factor.

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

Step 3.13.3

Rewrite the expression.

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

Step 3.13.4

Divide $- 3 v^{\frac{1}{2}}$ by $1$ .

$\frac{v (3 v^{2} - 3 v^{\frac{1}{2}}) - (v^{3} - 2 v \sqrt{v}) \frac{d}{d v} [v]}{v^{2}}$

Step 4

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

$\frac{v (3 v^{2} - 3 v^{\frac{1}{2}}) - (v^{3} - 2 v \sqrt{v}) \cdot 1}{v^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Simplify the numerator.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.4.1.2

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

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Step 5.4.1.2.1

Move $v^{2}$ .

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

Step 5.4.1.2.2

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

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Step 5.4.1.2.2.1

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

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

Step 5.4.1.2.2.2

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

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

Step 5.4.1.2.3

Add $2$ and $1$ .

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

Step 5.4.1.3

Rewrite using the commutative property of multiplication.

$\frac{3 v^{3} - 3 v \cdot v^{\frac{1}{2}} - v^{3} \cdot 1 - (- 2 v \sqrt{v}) \cdot 1}{v^{2}}$

Step 5.4.1.4

Multiply $v$ by $v^{\frac{1}{2}}$ by adding the exponents.

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Step 5.4.1.4.1

Move $v^{\frac{1}{2}}$ .

$\frac{3 v^{3} - 3 (v^{\frac{1}{2}} v) - v^{3} \cdot 1 - (- 2 v \sqrt{v}) \cdot 1}{v^{2}}$

Step 5.4.1.4.2

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

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Step 5.4.1.4.2.1

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

$\frac{3 v^{3} - 3 (v^{\frac{1}{2}} v^{1}) - v^{3} \cdot 1 - (- 2 v \sqrt{v}) \cdot 1}{v^{2}}$

Step 5.4.1.4.2.2

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

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

Step 5.4.1.4.3

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

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

Step 5.4.1.4.4

Combine the numerators over the common denominator.

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

Step 5.4.1.4.5

Add $1$ and $2$ .

$\frac{3 v^{3} - 3 v^{\frac{3}{2}} - v^{3} \cdot 1 - (- 2 v \sqrt{v}) \cdot 1}{v^{2}}$

Step 5.4.1.5

Multiply $- 1$ by $1$ .

$\frac{3 v^{3} - 3 v^{\frac{3}{2}} - v^{3} - (- 2 v \sqrt{v}) \cdot 1}{v^{2}}$

Step 5.4.1.6

Multiply $- 2$ by $- 1$ .

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

Step 5.4.1.7

Multiply $2$ by $1$ .

$\frac{3 v^{3} - 3 v^{\frac{3}{2}} - v^{3} + 2 v \sqrt{v}}{v^{2}}$

Step 5.4.2

Subtract $v^{3}$ from $3 v^{3}$ .

$\frac{2 v^{3} - 3 v^{\frac{3}{2}} + 2 v \sqrt{v}}{v^{2}}$

Step 5.5

Combine terms.

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{v}$ as $v^{\frac{1}{2}}$ .

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

Step 5.5.2

Multiply $v$ by $v^{\frac{1}{2}}$ by adding the exponents.

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Step 5.5.2.1

Move $v^{\frac{1}{2}}$ .

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

Step 5.5.2.2

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

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Step 5.5.2.2.1

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

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

Step 5.5.2.2.2

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

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

Step 5.5.2.3

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

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

Step 5.5.2.4

Combine the numerators over the common denominator.

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

Step 5.5.2.5

Add $1$ and $2$ .

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

Step 5.5.3

Add $- 3 v^{\frac{3}{2}}$ and $2 v^{\frac{3}{2}}$ .

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

Step 5.5.4

Factor $v^{\frac{3}{2}}$ out of $2 v^{3} - v^{\frac{3}{2}}$ .

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Step 5.5.4.1

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

$\frac{v^{\frac{3}{2}} (2 v^{\frac{3}{2}}) - v^{\frac{3}{2}}}{v^{2}}$

Step 5.5.4.2

Factor $v^{\frac{3}{2}}$ out of $- v^{\frac{3}{2}}$ .

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

Step 5.5.4.3

Factor $v^{\frac{3}{2}}$ out of $v^{\frac{3}{2}} (2 v^{\frac{3}{2}}) + v^{\frac{3}{2}} \cdot - 1$ .

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

Step 5.5.5

Move $v^{\frac{3}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{2 v^{\frac{3}{2}} - 1}{v^{2} v^{- \frac{3}{2}}}$

Step 5.5.6

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

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Step 5.5.6.1

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

$\frac{2 v^{\frac{3}{2}} - 1}{v^{2 - \frac{3}{2}}}$

Step 5.5.6.2

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

$\frac{2 v^{\frac{3}{2}} - 1}{v^{2 \cdot \frac{2}{2} - \frac{3}{2}}}$

Step 5.5.6.3

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

$\frac{2 v^{\frac{3}{2}} - 1}{v^{\frac{2 \cdot 2}{2} - \frac{3}{2}}}$

Step 5.5.6.4

Combine the numerators over the common denominator.

$\frac{2 v^{\frac{3}{2}} - 1}{v^{\frac{2 \cdot 2 - 3}{2}}}$

Step 5.5.6.5

Simplify the numerator.

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Step 5.5.6.5.1

Multiply $2$ by $2$ .

$\frac{2 v^{\frac{3}{2}} - 1}{v^{\frac{4 - 3}{2}}}$

Step 5.5.6.5.2

Subtract $3$ from $4$ .

$\frac{2 v^{\frac{3}{2}} - 1}{v^{\frac{1}{2}}}$