Calculus Examples

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

Step 1

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

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Add $1$ and $2$ .

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

$\frac{d}{d v} [\frac{v^{\frac{3}{2}} v^{\frac{3}{2}} + v^{\frac{3}{2}} \cdot - 2}{v}]$

Step 3.3

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

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

Step 4

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

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

Step 5

Multiply $v^{\frac{3}{2}}$ by $v^{- 1}$ by adding the exponents.

Tap for more steps...

Step 5.1

Move $v^{- 1}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Simplify the numerator.

Tap for more steps...

Step 5.6.1

Multiply $- 1$ by $2$ .

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

Step 5.6.2

Add $- 2$ and $3$ .

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

Step 6

Differentiate using the Product Rule which states that $\frac{d}{d v} [f (v) g (v)]$ is $f (v) \frac{d}{d v} [g (v)] + g (v) \frac{d}{d v} [f (v)]$ where $f (v) = v^{\frac{1}{2}}$ and $g (v) = v^{\frac{3}{2}} - 2$ .

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

Step 7

Differentiate.

Tap for more steps...

Step 7.1

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

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

Step 7.2

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

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

Step 8

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

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify the numerator.

Tap for more steps...

Step 11.1

Multiply $- 1$ by $2$ .

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

Step 11.2

Subtract $2$ from $3$ .

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

Step 12

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

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

Step 13

Since $- 2$ is constant with respect to $v$ , the derivative of $- 2$ with respect to $v$ is $0$ .

$v^{\frac{1}{2}} (\frac{3 v^{\frac{1}{2}}}{2} + 0) + (v^{\frac{3}{2}} - 2) \frac{d}{d v} [v^{\frac{1}{2}}]$

Step 14

Combine fractions.

Tap for more steps...

Step 14.1

Add $\frac{3 v^{\frac{1}{2}}}{2}$ and $0$ .

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

Step 14.2

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

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

Step 15

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

Tap for more steps...

Step 15.1

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

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

Step 15.2

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

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

Step 15.3

Combine the numerators over the common denominator.

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

Step 15.4

Add $1$ and $1$ .

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

Step 15.5

Divide $2$ by $2$ .

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

Step 16

Simplify the expression.

Tap for more steps...

Step 16.1

Simplify $v^{1} \cdot 3$ .

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

Step 16.2

Move $3$ to the left of $v$ .

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

Step 17

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

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

Step 18

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

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

Step 19

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

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

Step 20

Combine the numerators over the common denominator.

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

Step 21

Simplify the numerator.

Tap for more steps...

Step 21.1

Multiply $- 1$ by $2$ .

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

Step 21.2

Subtract $2$ from $1$ .

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

Step 22

Move the negative in front of the fraction.

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

Step 23

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

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

Step 24

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

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

Step 25

To write $(v^{\frac{3}{2}} - 2) \frac{1}{2 v^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 26

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

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

Step 27

Combine the numerators over the common denominator.

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

Step 28

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

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

Step 29

Cancel the common factor.

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

Step 30

Rewrite the expression.

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

Step 31

Simplify.

Tap for more steps...

Step 31.1

Apply the distributive property.

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

Step 31.2

Simplify the numerator.

Tap for more steps...

Step 31.2.1

Simplify each term.

Tap for more steps...

Step 31.2.1.1

Cancel the common factor of $v^{\frac{1}{2}}$ .

Tap for more steps...

Step 31.2.1.1.1

Factor $v^{\frac{1}{2}}$ out of $v^{\frac{3}{2}}$ .

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

Step 31.2.1.1.2

Cancel the common factor.

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

Step 31.2.1.1.3

Rewrite the expression.

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

Step 31.2.1.2

Divide $2$ by $2$ .

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

Step 31.2.1.3

Simplify.

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

Step 31.2.1.4

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

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

Step 31.2.1.5

Move the negative in front of the fraction.

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

Step 31.2.2

Add $3 v$ and $v$ .

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

Step 31.3

Combine terms.

Tap for more steps...

Step 31.3.1

Factor $2$ out of $4 v$ .

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

Step 31.3.2

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

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

Step 31.3.3

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

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

Step 31.3.4

Cancel the common factors.

Tap for more steps...

Step 31.3.4.1

Factor $2$ out of $2$ .

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

Step 31.3.4.2

Cancel the common factor.

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

Step 31.3.4.3

Rewrite the expression.

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

Step 31.3.4.4

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

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