Calculus Examples

$p = (\frac{q^{8} + 4}{2 q}) (\frac{q^{7} + 6}{q})$

Step 1

Multiply $\frac{q^{8} + 4}{2 q}$ by $\frac{q^{7} + 6}{q}$ .

$p = \frac{q^{8} + 4}{2 q} \cdot (\frac{q^{7} + 6}{q})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d q} (p) = \frac{d}{d q} (\frac{q^{8} + 4}{2 q} \cdot (\frac{q^{7} + 6}{q}))$

Step 3

The derivative of $p$ with respect to $q$ is $p'$ .

$p'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Multiply $\frac{q^{8} + 4}{2 q}$ by $\frac{q^{7} + 6}{q}$ .

$\frac{d}{d q} [\frac{(q^{8} + 4) (q^{7} + 6)}{2 q \cdot q}]$

Step 4.2

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

$\frac{d}{d q} [\frac{(q^{8} + 4) (q^{7} + 6)}{2 (q^{1} q)}]$

Step 4.3

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

$\frac{d}{d q} [\frac{(q^{8} + 4) (q^{7} + 6)}{2 (q^{1} q^{1})}]$

Step 4.4

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

$\frac{d}{d q} [\frac{(q^{8} + 4) (q^{7} + 6)}{2 q^{1 + 1}}]$

Step 4.5

Differentiate using the Constant Multiple Rule.

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Step 4.5.1

Add $1$ and $1$ .

$\frac{d}{d q} [\frac{(q^{8} + 4) (q^{7} + 6)}{2 q^{2}}]$

Step 4.5.2

Since $\frac{1}{2}$ is constant with respect to $q$ , the derivative of $\frac{(q^{8} + 4) (q^{7} + 6)}{2 q^{2}}$ with respect to $q$ is $\frac{1}{2} \frac{d}{d q} [\frac{(q^{8} + 4) (q^{7} + 6)}{q^{2}}]$ .

$\frac{1}{2} \frac{d}{d q} [\frac{(q^{8} + 4) (q^{7} + 6)}{q^{2}}]$

Step 4.6

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 $f (q) = (q^{8} + 4) (q^{7} + 6)$ and $g (q) = q^{2}$ .

$\frac{1}{2} \cdot \frac{q^{2} \frac{d}{d q} [(q^{8} + 4) (q^{7} + 6)] - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{{(q^{2})}^{2}}$

Step 4.7

Multiply the exponents in ${(q^{2})}^{2}$ .

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Step 4.7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} \cdot \frac{q^{2} \frac{d}{d q} [(q^{8} + 4) (q^{7} + 6)] - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{2 \cdot 2}}$

Step 4.7.2

Multiply $2$ by $2$ .

$\frac{1}{2} \cdot \frac{q^{2} \frac{d}{d q} [(q^{8} + 4) (q^{7} + 6)] - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.8

Differentiate using the Product Rule which states that $\frac{d}{d q} [f (q) g (q)]$ is $f (q) \frac{d}{d q} [g (q)] + g (q) \frac{d}{d q} [f (q)]$ where $f (q) = q^{8} + 4$ and $g (q) = q^{7} + 6$ .

$\frac{1}{2} \cdot \frac{q^{2} ((q^{8} + 4) \frac{d}{d q} [q^{7} + 6] + (q^{7} + 6) \frac{d}{d q} [q^{8} + 4]) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9

Differentiate.

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Step 4.9.1

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

$\frac{1}{2} \cdot \frac{q^{2} ((q^{8} + 4) (\frac{d}{d q} [q^{7}] + \frac{d}{d q} [6]) + (q^{7} + 6) \frac{d}{d q} [q^{8} + 4]) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.2

Differentiate using the Power Rule which states that $\frac{d}{d q} [q^{n}]$ is $n q^{n - 1}$ where $n = 7$ .

$\frac{1}{2} \cdot \frac{q^{2} ((q^{8} + 4) (7 q^{6} + \frac{d}{d q} [6]) + (q^{7} + 6) \frac{d}{d q} [q^{8} + 4]) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.3

Since $6$ is constant with respect to $q$ , the derivative of $6$ with respect to $q$ is $0$ .

$\frac{1}{2} \cdot \frac{q^{2} ((q^{8} + 4) (7 q^{6} + 0) + (q^{7} + 6) \frac{d}{d q} [q^{8} + 4]) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.4

Simplify the expression.

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Step 4.9.4.1

Add $7 q^{6}$ and $0$ .

$\frac{1}{2} \cdot \frac{q^{2} ((q^{8} + 4) (7 q^{6}) + (q^{7} + 6) \frac{d}{d q} [q^{8} + 4]) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.4.2

Move $7$ to the left of $q^{8} + 4$ .

$\frac{1}{2} \cdot \frac{q^{2} (7 \cdot (q^{8} + 4) q^{6} + (q^{7} + 6) \frac{d}{d q} [q^{8} + 4]) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.5

By the Sum Rule, the derivative of $q^{8} + 4$ with respect to $q$ is $\frac{d}{d q} [q^{8}] + \frac{d}{d q} [4]$ .

$\frac{1}{2} \cdot \frac{q^{2} (7 (q^{8} + 4) q^{6} + (q^{7} + 6) (\frac{d}{d q} [q^{8}] + \frac{d}{d q} [4])) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.6

Differentiate using the Power Rule which states that $\frac{d}{d q} [q^{n}]$ is $n q^{n - 1}$ where $n = 8$ .

$\frac{1}{2} \cdot \frac{q^{2} (7 (q^{8} + 4) q^{6} + (q^{7} + 6) (8 q^{7} + \frac{d}{d q} [4])) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.7

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

$\frac{1}{2} \cdot \frac{q^{2} (7 (q^{8} + 4) q^{6} + (q^{7} + 6) (8 q^{7} + 0)) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.8

Simplify the expression.

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Step 4.9.8.1

Add $8 q^{7}$ and $0$ .

$\frac{1}{2} \cdot \frac{q^{2} (7 (q^{8} + 4) q^{6} + (q^{7} + 6) (8 q^{7})) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.8.2

Move $8$ to the left of $q^{7} + 6$ .

$\frac{1}{2} \cdot \frac{q^{2} (7 (q^{8} + 4) q^{6} + 8 \cdot (q^{7} + 6) q^{7}) - (q^{8} + 4) (q^{7} + 6) \frac{d}{d q} [q^{2}]}{q^{4}}$

Step 4.9.9

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

$\frac{1}{2} \cdot \frac{q^{2} (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7}) - (q^{8} + 4) (q^{7} + 6) (2 q)}{q^{4}}$

Step 4.9.10

Simplify with factoring out.

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Step 4.9.10.1

Multiply $2$ by $- 1$ .

$\frac{1}{2} \cdot \frac{q^{2} (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6) q}{q^{4}}$

Step 4.9.10.2

Factor $q$ out of $q^{2} (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6) q$ .

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Step 4.9.10.2.1

Factor $q$ out of $q^{2} (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7})$ .

$\frac{1}{2} \cdot \frac{q (q (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7})) - 2 (q^{8} + 4) (q^{7} + 6) q}{q^{4}}$

Step 4.9.10.2.2

Factor $q$ out of $- 2 (q^{8} + 4) (q^{7} + 6) q$ .

$\frac{1}{2} \cdot \frac{q (q (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7})) + q (- 2 (q^{8} + 4) (q^{7} + 6))}{q^{4}}$

Step 4.9.10.2.3

Factor $q$ out of $q (q (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7})) + q (- 2 (q^{8} + 4) (q^{7} + 6))$ .

$\frac{1}{2} \cdot \frac{q (q (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6))}{q^{4}}$

Step 4.10

Cancel the common factors.

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Step 4.10.1

Factor $q$ out of $q^{4}$ .

$\frac{1}{2} \cdot \frac{q (q (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6))}{q \cdot q^{3}}$

Step 4.10.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{q (q (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6))}{q \cdot q^{3}}$

Step 4.10.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{q (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6)}{q^{3}}$

Step 4.11

Multiply $\frac{1}{2}$ by $\frac{q (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6)}{q^{3}}$ .

$\frac{q (7 (q^{8} + 4) q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12

Simplify.

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Step 4.12.1

Apply the distributive property.

$\frac{q ((7 q^{8} + 7 \cdot 4) q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.2

Apply the distributive property.

$\frac{q (7 q^{8} q^{6} + 7 \cdot 4 q^{6} + 8 (q^{7} + 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.3

Apply the distributive property.

$\frac{q (7 q^{8} q^{6} + 7 \cdot 4 q^{6} + (8 q^{7} + 8 \cdot 6) q^{7}) - 2 (q^{8} + 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.4

Apply the distributive property.

$\frac{q (7 q^{8} q^{6} + 7 \cdot 4 q^{6} + 8 q^{7} q^{7} + 8 \cdot 6 q^{7}) - 2 (q^{8} + 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.5

Apply the distributive property.

$\frac{q (7 q^{8} q^{6}) + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) - 2 (q^{8} + 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.6

Apply the distributive property.

$\frac{q (7 q^{8} q^{6}) + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7

Simplify the numerator.

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Step 4.12.7.1

Simplify each term.

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Step 4.12.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{7 q (q^{8} q^{6}) + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.2

Multiply $q$ by $q^{8}$ by adding the exponents.

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Step 4.12.7.1.2.1

Move $q^{8}$ .

$\frac{7 (q^{8} q) q^{6} + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.2.2

Multiply $q^{8}$ by $q$ .

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Step 4.12.7.1.2.2.1

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

$\frac{7 (q^{8} q^{1}) q^{6} + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.2.2.2

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

$\frac{7 q^{8 + 1} q^{6} + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.2.3

Add $8$ and $1$ .

$\frac{7 q^{9} q^{6} + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.3

Multiply $q^{9}$ by $q^{6}$ by adding the exponents.

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Step 4.12.7.1.3.1

Move $q^{6}$ .

$\frac{7 (q^{6} q^{9}) + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.3.2

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

$\frac{7 q^{6 + 9} + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.3.3

Add $6$ and $9$ .

$\frac{7 q^{15} + q (7 \cdot 4 q^{6}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.4

Rewrite using the commutative property of multiplication.

$\frac{7 q^{15} + 7 \cdot 4 q \cdot q^{6} + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.5

Multiply $q$ by $q^{6}$ by adding the exponents.

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Step 4.12.7.1.5.1

Move $q^{6}$ .

$\frac{7 q^{15} + 7 \cdot 4 (q^{6} q) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.5.2

Multiply $q^{6}$ by $q$ .

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Step 4.12.7.1.5.2.1

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

$\frac{7 q^{15} + 7 \cdot 4 (q^{6} q^{1}) + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.5.2.2

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

$\frac{7 q^{15} + 7 \cdot 4 q^{6 + 1} + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.5.3

Add $6$ and $1$ .

$\frac{7 q^{15} + 7 \cdot 4 q^{7} + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.6

Multiply $7$ by $4$ .

$\frac{7 q^{15} + 28 q^{7} + q (8 q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.7

Rewrite using the commutative property of multiplication.

$\frac{7 q^{15} + 28 q^{7} + 8 q (q^{7} q^{7}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.8

Multiply $q$ by $q^{7}$ by adding the exponents.

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Step 4.12.7.1.8.1

Move $q^{7}$ .

$\frac{7 q^{15} + 28 q^{7} + 8 (q^{7} q) q^{7} + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.8.2

Multiply $q^{7}$ by $q$ .

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Step 4.12.7.1.8.2.1

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

$\frac{7 q^{15} + 28 q^{7} + 8 (q^{7} q^{1}) q^{7} + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.8.2.2

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

$\frac{7 q^{15} + 28 q^{7} + 8 q^{7 + 1} q^{7} + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.8.3

Add $7$ and $1$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{8} q^{7} + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.9

Multiply $q^{8}$ by $q^{7}$ by adding the exponents.

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Step 4.12.7.1.9.1

Move $q^{7}$ .

$\frac{7 q^{15} + 28 q^{7} + 8 (q^{7} q^{8}) + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.9.2

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

$\frac{7 q^{15} + 28 q^{7} + 8 q^{7 + 8} + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.9.3

Add $7$ and $8$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + q (8 \cdot 6 q^{7}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.10

Rewrite using the commutative property of multiplication.

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 8 \cdot 6 q \cdot q^{7} + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.11

Multiply $q$ by $q^{7}$ by adding the exponents.

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Step 4.12.7.1.11.1

Move $q^{7}$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 8 \cdot 6 (q^{7} q) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.11.2

Multiply $q^{7}$ by $q$ .

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Step 4.12.7.1.11.2.1

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

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 8 \cdot 6 (q^{7} q^{1}) + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.11.2.2

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

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 8 \cdot 6 q^{7 + 1} + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.11.3

Add $7$ and $1$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 8 \cdot 6 q^{8} + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.12

Multiply $8$ by $6$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} + (- 2 q^{8} - 2 \cdot 4) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.13

Multiply $- 2$ by $4$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} + (- 2 q^{8} - 8) (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.14

Expand $(- 2 q^{8} - 8) (q^{7} + 6)$ using the FOIL Method.

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Step 4.12.7.1.14.1

Apply the distributive property.

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} - 2 q^{8} (q^{7} + 6) - 8 (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.14.2

Apply the distributive property.

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} - 2 q^{8} q^{7} - 2 q^{8} \cdot 6 - 8 (q^{7} + 6)}{2 q^{3}}$

Step 4.12.7.1.14.3

Apply the distributive property.

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} - 2 q^{8} q^{7} - 2 q^{8} \cdot 6 - 8 q^{7} - 8 \cdot 6}{2 q^{3}}$

Step 4.12.7.1.15

Simplify each term.

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Step 4.12.7.1.15.1

Multiply $q^{8}$ by $q^{7}$ by adding the exponents.

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Step 4.12.7.1.15.1.1

Move $q^{7}$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} - 2 (q^{7} q^{8}) - 2 q^{8} \cdot 6 - 8 q^{7} - 8 \cdot 6}{2 q^{3}}$

Step 4.12.7.1.15.1.2

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

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} - 2 q^{7 + 8} - 2 q^{8} \cdot 6 - 8 q^{7} - 8 \cdot 6}{2 q^{3}}$

Step 4.12.7.1.15.1.3

Add $7$ and $8$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} - 2 q^{15} - 2 q^{8} \cdot 6 - 8 q^{7} - 8 \cdot 6}{2 q^{3}}$

Step 4.12.7.1.15.2

Multiply $6$ by $- 2$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} - 2 q^{15} - 12 q^{8} - 8 q^{7} - 8 \cdot 6}{2 q^{3}}$

Step 4.12.7.1.15.3

Multiply $- 8$ by $6$ .

$\frac{7 q^{15} + 28 q^{7} + 8 q^{15} + 48 q^{8} - 2 q^{15} - 12 q^{8} - 8 q^{7} - 48}{2 q^{3}}$

Step 4.12.7.2

Add $7 q^{15}$ and $8 q^{15}$ .

$\frac{15 q^{15} + 28 q^{7} + 48 q^{8} - 2 q^{15} - 12 q^{8} - 8 q^{7} - 48}{2 q^{3}}$

Step 4.12.7.3

Subtract $2 q^{15}$ from $15 q^{15}$ .

$\frac{13 q^{15} + 28 q^{7} + 48 q^{8} - 12 q^{8} - 8 q^{7} - 48}{2 q^{3}}$

Step 4.12.7.4

Subtract $8 q^{7}$ from $28 q^{7}$ .

$\frac{13 q^{15} + 48 q^{8} - 12 q^{8} + 20 q^{7} - 48}{2 q^{3}}$

Step 4.12.7.5

Subtract $12 q^{8}$ from $48 q^{8}$ .

$\frac{13 q^{15} + 36 q^{8} + 20 q^{7} - 48}{2 q^{3}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$p' = \frac{13 q^{15} + 36 q^{8} + 20 q^{7} - 48}{2 q^{3}}$

Step 6

Replace $p'$ with $\frac{d p}{d q}$ .

$\frac{d p}{d q} = \frac{13 q^{15} + 36 q^{8} + 20 q^{7} - 48}{2 q^{3}}$