Calculus Examples

$y = \frac{4 - v^{2}}{4 + v^{2}}$

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 $f (v) = 4 - v^{2}$ and $g (v) = 4 + v^{2}$ .

$\frac{(4 + v^{2}) \frac{d}{d v} [4 - v^{2}] - (4 - v^{2}) \frac{d}{d v} [4 + v^{2}]}{{(4 + v^{2})}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 - v^{2}$ with respect to $v$ is $\frac{d}{d v} [4] + \frac{d}{d v} [- v^{2}]$ .

$\frac{(4 + v^{2}) (\frac{d}{d v} [4] + \frac{d}{d v} [- v^{2}]) - (4 - v^{2}) \frac{d}{d v} [4 + v^{2}]}{{(4 + v^{2})}^{2}}$

Step 2.2

Since $4$ is constant with respect to $v$ , the derivative of $4$ with respect to $v$ is $0$ .

$\frac{(4 + v^{2}) (0 + \frac{d}{d v} [- v^{2}]) - (4 - v^{2}) \frac{d}{d v} [4 + v^{2}]}{{(4 + v^{2})}^{2}}$

Step 2.3

Add $0$ and $\frac{d}{d v} [- v^{2}]$ .

$\frac{(4 + v^{2}) \frac{d}{d v} [- v^{2}] - (4 - v^{2}) \frac{d}{d v} [4 + v^{2}]}{{(4 + v^{2})}^{2}}$

Step 2.4

Since $- 1$ is constant with respect to $v$ , the derivative of $- v^{2}$ with respect to $v$ is $- \frac{d}{d v} [v^{2}]$ .

$\frac{(4 + v^{2}) (- \frac{d}{d v} [v^{2}]) - (4 - v^{2}) \frac{d}{d v} [4 + v^{2}]}{{(4 + v^{2})}^{2}}$

Step 2.5

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

$\frac{(4 + v^{2}) (- (2 v)) - (4 - v^{2}) \frac{d}{d v} [4 + v^{2}]}{{(4 + v^{2})}^{2}}$

Step 2.6

Multiply $2$ by $- 1$ .

$\frac{(4 + v^{2}) (- 2 v) - (4 - v^{2}) \frac{d}{d v} [4 + v^{2}]}{{(4 + v^{2})}^{2}}$

Step 2.7

By the Sum Rule, the derivative of $4 + v^{2}$ with respect to $v$ is $\frac{d}{d v} [4] + \frac{d}{d v} [v^{2}]$ .

$\frac{(4 + v^{2}) (- 2 v) - (4 - v^{2}) (\frac{d}{d v} [4] + \frac{d}{d v} [v^{2}])}{{(4 + v^{2})}^{2}}$

Step 2.8

Since $4$ is constant with respect to $v$ , the derivative of $4$ with respect to $v$ is $0$ .

$\frac{(4 + v^{2}) (- 2 v) - (4 - v^{2}) (0 + \frac{d}{d v} [v^{2}])}{{(4 + v^{2})}^{2}}$

Step 2.9

Add $0$ and $\frac{d}{d v} [v^{2}]$ .

$\frac{(4 + v^{2}) (- 2 v) - (4 - v^{2}) \frac{d}{d v} [v^{2}]}{{(4 + v^{2})}^{2}}$

Step 2.10

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

$\frac{(4 + v^{2}) (- 2 v) - (4 - v^{2}) (2 v)}{{(4 + v^{2})}^{2}}$

Step 2.11

Multiply $2$ by $- 1$ .

$\frac{(4 + v^{2}) (- 2 v) - 2 (4 - v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{4 (- 2 v) + v^{2} (- 2 v) - 2 (4 - v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{4 (- 2 v) + v^{2} (- 2 v) + (- 2 \cdot 4 - 2 (- v^{2})) v}{{(4 + v^{2})}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{4 (- 2 v) + v^{2} (- 2 v) - 2 \cdot 4 v - 2 (- v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $- 2$ by $4$ .

$\frac{- 8 v + v^{2} (- 2 v) - 2 \cdot 4 v - 2 (- v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3.4.1.2

Rewrite using the commutative property of multiplication.

$\frac{- 8 v - 2 v^{2} v - 2 \cdot 4 v - 2 (- v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3.4.1.3

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

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Step 3.4.1.3.1

Move $v$ .

$\frac{- 8 v - 2 (v \cdot v^{2}) - 2 \cdot 4 v - 2 (- v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3.4.1.3.2

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

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Step 3.4.1.3.2.1

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

$\frac{- 8 v - 2 (v^{1} v^{2}) - 2 \cdot 4 v - 2 (- v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3.4.1.3.2.2

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

$\frac{- 8 v - 2 v^{1 + 2} - 2 \cdot 4 v - 2 (- v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3.4.1.3.3

Add $1$ and $2$ .

$\frac{- 8 v - 2 v^{3} - 2 \cdot 4 v - 2 (- v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3.4.1.4

Multiply $- 2$ by $4$ .

$\frac{- 8 v - 2 v^{3} - 8 v - 2 (- v^{2}) v}{{(4 + v^{2})}^{2}}$

Step 3.4.1.5

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

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Step 3.4.1.5.1

Move $v$ .

$\frac{- 8 v - 2 v^{3} - 8 v - 2 (- (v \cdot v^{2}))}{{(4 + v^{2})}^{2}}$

Step 3.4.1.5.2

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

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Step 3.4.1.5.2.1

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

$\frac{- 8 v - 2 v^{3} - 8 v - 2 (- (v^{1} v^{2}))}{{(4 + v^{2})}^{2}}$

Step 3.4.1.5.2.2

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

$\frac{- 8 v - 2 v^{3} - 8 v - 2 (- v^{1 + 2})}{{(4 + v^{2})}^{2}}$

Step 3.4.1.5.3

Add $1$ and $2$ .

$\frac{- 8 v - 2 v^{3} - 8 v - 2 (- v^{3})}{{(4 + v^{2})}^{2}}$

Step 3.4.1.6

Multiply $- 1$ by $- 2$ .

$\frac{- 8 v - 2 v^{3} - 8 v + 2 v^{3}}{{(4 + v^{2})}^{2}}$

Step 3.4.2

Combine the opposite terms in $- 8 v - 2 v^{3} - 8 v + 2 v^{3}$ .

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Step 3.4.2.1

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

$\frac{- 8 v - 8 v + 0}{{(4 + v^{2})}^{2}}$

Step 3.4.2.2

Add $- 8 v - 8 v$ and $0$ .

$\frac{- 8 v - 8 v}{{(4 + v^{2})}^{2}}$

Step 3.4.3

Subtract $8 v$ from $- 8 v$ .

$\frac{- 16 v}{{(4 + v^{2})}^{2}}$

Step 3.5

Move the negative in front of the fraction.

$- \frac{16 v}{{(4 + v^{2})}^{2}}$