Precalculus Examples

$(\frac{102 - 3 q^{2}}{q + 2})$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d q} [\frac{f (q)}{g (q)}]$ is $\frac{g (q) \frac{d}{d q} [f (q)] - f (q) \frac{d}{d q} [g (q)]}{{g (q)}^{2}}$ where $f (q) = 102 - 3 q^{2}$ and $g (q) = q + 2$ .

$\frac{(q + 2) \frac{d}{d q} [102 - 3 q^{2}] - (102 - 3 q^{2}) \frac{d}{d q} [q + 2]}{{(q + 2)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $102 - 3 q^{2}$ with respect to $q$ is $\frac{d}{d q} [102] + \frac{d}{d q} [- 3 q^{2}]$ .

$\frac{(q + 2) (\frac{d}{d q} [102] + \frac{d}{d q} [- 3 q^{2}]) - (102 - 3 q^{2}) \frac{d}{d q} [q + 2]}{{(q + 2)}^{2}}$

Step 2.2

Since $102$ is constant with respect to $q$ , the derivative of $102$ with respect to $q$ is $0$ .

$\frac{(q + 2) (0 + \frac{d}{d q} [- 3 q^{2}]) - (102 - 3 q^{2}) \frac{d}{d q} [q + 2]}{{(q + 2)}^{2}}$

Step 2.3

Add $0$ and $\frac{d}{d q} [- 3 q^{2}]$ .

$\frac{(q + 2) \frac{d}{d q} [- 3 q^{2}] - (102 - 3 q^{2}) \frac{d}{d q} [q + 2]}{{(q + 2)}^{2}}$

Step 2.4

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

$\frac{(q + 2) (- 3 \frac{d}{d q} [q^{2}]) - (102 - 3 q^{2}) \frac{d}{d q} [q + 2]}{{(q + 2)}^{2}}$

Step 2.5

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

$\frac{(q + 2) (- 3 (2 q)) - (102 - 3 q^{2}) \frac{d}{d q} [q + 2]}{{(q + 2)}^{2}}$

Step 2.6

Multiply $2$ by $- 3$ .

$\frac{(q + 2) (- 6 q) - (102 - 3 q^{2}) \frac{d}{d q} [q + 2]}{{(q + 2)}^{2}}$

Step 2.7

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

$\frac{(q + 2) (- 6 q) - (102 - 3 q^{2}) (\frac{d}{d q} [q] + \frac{d}{d q} [2])}{{(q + 2)}^{2}}$

Step 2.8

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

$\frac{(q + 2) (- 6 q) - (102 - 3 q^{2}) (1 + \frac{d}{d q} [2])}{{(q + 2)}^{2}}$

Step 2.9

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

$\frac{(q + 2) (- 6 q) - (102 - 3 q^{2}) (1 + 0)}{{(q + 2)}^{2}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $1$ and $0$ .

$\frac{(q + 2) (- 6 q) - (102 - 3 q^{2}) \cdot 1}{{(q + 2)}^{2}}$

Step 2.10.2

Multiply $- 1$ by $1$ .

$\frac{(q + 2) (- 6 q) - (102 - 3 q^{2})}{{(q + 2)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{q (- 6 q) + 2 (- 6 q) - (102 - 3 q^{2})}{{(q + 2)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{q (- 6 q) + 2 (- 6 q) - 1 \cdot 102 - (- 3 q^{2})}{{(q + 2)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 6 q \cdot q + 2 (- 6 q) - 1 \cdot 102 - (- 3 q^{2})}{{(q + 2)}^{2}}$

Step 3.3.1.2

Multiply $q$ by $q$ by adding the exponents.

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Step 3.3.1.2.1

Move $q$ .

$\frac{- 6 (q \cdot q) + 2 (- 6 q) - 1 \cdot 102 - (- 3 q^{2})}{{(q + 2)}^{2}}$

Step 3.3.1.2.2

Multiply $q$ by $q$ .

$\frac{- 6 q^{2} + 2 (- 6 q) - 1 \cdot 102 - (- 3 q^{2})}{{(q + 2)}^{2}}$

Step 3.3.1.3

Multiply $- 6$ by $2$ .

$\frac{- 6 q^{2} - 12 q - 1 \cdot 102 - (- 3 q^{2})}{{(q + 2)}^{2}}$

Step 3.3.1.4

Multiply $- 1$ by $102$ .

$\frac{- 6 q^{2} - 12 q - 102 - (- 3 q^{2})}{{(q + 2)}^{2}}$

Step 3.3.1.5

Multiply $- 3$ by $- 1$ .

$\frac{- 6 q^{2} - 12 q - 102 + 3 q^{2}}{{(q + 2)}^{2}}$

Step 3.3.2

Add $- 6 q^{2}$ and $3 q^{2}$ .

$\frac{- 3 q^{2} - 12 q - 102}{{(q + 2)}^{2}}$

Step 3.4

Factor $3$ out of $- 3 q^{2} - 12 q - 102$ .

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Step 3.4.1

Factor $3$ out of $- 3 q^{2}$ .

$\frac{3 (- q^{2}) - 12 q - 102}{{(q + 2)}^{2}}$

Step 3.4.2

Factor $3$ out of $- 12 q$ .

$\frac{3 (- q^{2}) + 3 (- 4 q) - 102}{{(q + 2)}^{2}}$

Step 3.4.3

Factor $3$ out of $- 102$ .

$\frac{3 (- q^{2}) + 3 (- 4 q) + 3 (- 34)}{{(q + 2)}^{2}}$

Step 3.4.4

Factor $3$ out of $3 (- q^{2}) + 3 (- 4 q)$ .

$\frac{3 (- q^{2} - 4 q) + 3 (- 34)}{{(q + 2)}^{2}}$

Step 3.4.5

Factor $3$ out of $3 (- q^{2} - 4 q) + 3 (- 34)$ .

$\frac{3 (- q^{2} - 4 q - 34)}{{(q + 2)}^{2}}$

Step 3.5

Factor $- 1$ out of $- q^{2}$ .

$\frac{3 (- (q^{2}) - 4 q - 34)}{{(q + 2)}^{2}}$

Step 3.6

Factor $- 1$ out of $- 4 q$ .

$\frac{3 (- (q^{2}) - (4 q) - 34)}{{(q + 2)}^{2}}$

Step 3.7

Factor $- 1$ out of $- (q^{2}) - (4 q)$ .

$\frac{3 (- (q^{2} + 4 q) - 34)}{{(q + 2)}^{2}}$

Step 3.8

Rewrite $- 34$ as $- 1 (34)$ .

$\frac{3 (- (q^{2} + 4 q) - 1 (34))}{{(q + 2)}^{2}}$

Step 3.9

Factor $- 1$ out of $- (q^{2} + 4 q) - 1 (34)$ .

$\frac{3 (- (q^{2} + 4 q + 34))}{{(q + 2)}^{2}}$

Step 3.10

Rewrite $- (q^{2} + 4 q + 34)$ as $- 1 (q^{2} + 4 q + 34)$ .

$\frac{3 (- 1 (q^{2} + 4 q + 34))}{{(q + 2)}^{2}}$

Step 3.11

Move the negative in front of the fraction.

$- \frac{3 (q^{2} + 4 q + 34)}{{(q + 2)}^{2}}$