Calculus Examples

$p = \frac{6 \sqrt[6]{q}}{1 - q}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{q}$ as $q^{\frac{1}{6}}$ .

$p = \frac{6 q^{\frac{1}{6}}}{1 - q}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d q} (p) = \frac{d}{d q} (\frac{6 q^{\frac{1}{6}}}{1 - q})$

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

Since $6$ is constant with respect to $q$ , the derivative of $\frac{6 q^{\frac{1}{6}}}{1 - q}$ with respect to $q$ is $6 \frac{d}{d q} [\frac{q^{\frac{1}{6}}}{1 - q}]$ .

$6 \frac{d}{d q} [\frac{q^{\frac{1}{6}}}{1 - q}]$

Step 4.2

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 $f (q) = q^{\frac{1}{6}}$ and $g (q) = 1 - q$ .

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

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d q} [q^{n}]$ is $n q^{n - 1}$ where $n = \frac{1}{6}$ .

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

Step 4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 4.5

Combine $- 1$ and $\frac{6}{6}$ .

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $6$ .

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

Step 4.7.2

Subtract $6$ from $1$ .

$6 \frac{(1 - q) (\frac{1}{6} q^{\frac{- 5}{6}}) - q^{\frac{1}{6}} \frac{d}{d q} [1 - q]}{{(1 - q)}^{2}}$

Step 4.8

Combine fractions.

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Step 4.8.1

Move the negative in front of the fraction.

$6 \frac{(1 - q) (\frac{1}{6} q^{- \frac{5}{6}}) - q^{\frac{1}{6}} \frac{d}{d q} [1 - q]}{{(1 - q)}^{2}}$

Step 4.8.2

Combine $\frac{1}{6}$ and $q^{- \frac{5}{6}}$ .

$6 \frac{(1 - q) \frac{q^{- \frac{5}{6}}}{6} - q^{\frac{1}{6}} \frac{d}{d q} [1 - q]}{{(1 - q)}^{2}}$

Step 4.8.3

Move $q^{- \frac{5}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 \frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} - q^{\frac{1}{6}} \frac{d}{d q} [1 - q]}{{(1 - q)}^{2}}$

Step 4.9

By the Sum Rule, the derivative of $1 - q$ with respect to $q$ is $\frac{d}{d q} [1] + \frac{d}{d q} [- q]$ .

$6 \frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} - q^{\frac{1}{6}} (\frac{d}{d q} [1] + \frac{d}{d q} [- q])}{{(1 - q)}^{2}}$

Step 4.10

Since $1$ is constant with respect to $q$ , the derivative of $1$ with respect to $q$ is $0$ .

$6 \frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} - q^{\frac{1}{6}} (0 + \frac{d}{d q} [- q])}{{(1 - q)}^{2}}$

Step 4.11

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

$6 \frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} - q^{\frac{1}{6}} \frac{d}{d q} [- q]}{{(1 - q)}^{2}}$

Step 4.12

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

$6 \frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} - q^{\frac{1}{6}} (- \frac{d}{d q} [q])}{{(1 - q)}^{2}}$

Step 4.13

Multiply.

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Step 4.13.1

Multiply $- 1$ by $- 1$ .

$6 \frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} + 1 q^{\frac{1}{6}} \frac{d}{d q} [q]}{{(1 - q)}^{2}}$

Step 4.13.2

Multiply $q^{\frac{1}{6}}$ by $1$ .

$6 \frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} + q^{\frac{1}{6}} \frac{d}{d q} [q]}{{(1 - q)}^{2}}$

Step 4.14

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

$6 \frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} + q^{\frac{1}{6}} \cdot 1}{{(1 - q)}^{2}}$

Step 4.15

Combine fractions.

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Step 4.15.1

Multiply $q^{\frac{1}{6}}$ by $1$ .

$6 \frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} + q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.15.2

Combine $6$ and $\frac{(1 - q) \frac{1}{6 q^{\frac{5}{6}}} + q^{\frac{1}{6}}}{{(1 - q)}^{2}}$ .

$\frac{6 ((1 - q) \frac{1}{6 q^{\frac{5}{6}}} + q^{\frac{1}{6}})}{{(1 - q)}^{2}}$

Step 4.16

Simplify.

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Step 4.16.1

Apply the distributive property.

$\frac{6 (1 \frac{1}{6 q^{\frac{5}{6}}} - q \frac{1}{6 q^{\frac{5}{6}}} + q^{\frac{1}{6}})}{{(1 - q)}^{2}}$

Step 4.16.2

Apply the distributive property.

$\frac{6 (1 \frac{1}{6 q^{\frac{5}{6}}}) + 6 (- q \frac{1}{6 q^{\frac{5}{6}}}) + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3

Simplify the numerator.

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Step 4.16.3.1

Simplify each term.

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Step 4.16.3.1.1

Multiply $\frac{1}{6 q^{\frac{5}{6}}}$ by $1$ .

$\frac{6 \frac{1}{6 q^{\frac{5}{6}}} + 6 (- q \frac{1}{6 q^{\frac{5}{6}}}) + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.2

Cancel the common factor of $6$ .

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Step 4.16.3.1.2.1

Cancel the common factor.

$\frac{6 \frac{1}{6 q^{\frac{5}{6}}} + 6 (- q \frac{1}{6 q^{\frac{5}{6}}}) + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.2.2

Rewrite the expression.

$\frac{\frac{1}{q^{\frac{5}{6}}} + 6 (- q \frac{1}{6 q^{\frac{5}{6}}}) + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.3

Combine $\frac{1}{6 q^{\frac{5}{6}}}$ and $q$ .

$\frac{\frac{1}{q^{\frac{5}{6}}} + 6 (- \frac{q}{6 q^{\frac{5}{6}}}) + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.4

Cancel the common factor of $6$ .

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Step 4.16.3.1.4.1

Move the leading negative in $- \frac{q}{6 q^{\frac{5}{6}}}$ into the numerator.

$\frac{\frac{1}{q^{\frac{5}{6}}} + 6 \frac{- q}{6 q^{\frac{5}{6}}} + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.4.2

Cancel the common factor.

$\frac{\frac{1}{q^{\frac{5}{6}}} + 6 \frac{- q}{6 q^{\frac{5}{6}}} + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.4.3

Rewrite the expression.

$\frac{\frac{1}{q^{\frac{5}{6}}} + \frac{- q}{q^{\frac{5}{6}}} + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.5

Move $q^{\frac{5}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{\frac{1}{q^{\frac{5}{6}}} - q \cdot q^{- \frac{5}{6}} + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.6

Multiply $q$ by $q^{- \frac{5}{6}}$ by adding the exponents.

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Step 4.16.3.1.6.1

Move $q^{- \frac{5}{6}}$ .

$\frac{\frac{1}{q^{\frac{5}{6}}} - (q^{- \frac{5}{6}} q) + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.6.2

Multiply $q^{- \frac{5}{6}}$ by $q$ .

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Step 4.16.3.1.6.2.1

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

$\frac{\frac{1}{q^{\frac{5}{6}}} - (q^{- \frac{5}{6}} q^{1}) + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.6.2.2

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

$\frac{\frac{1}{q^{\frac{5}{6}}} - q^{- \frac{5}{6} + 1} + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.6.3

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

$\frac{\frac{1}{q^{\frac{5}{6}}} - q^{- \frac{5}{6} + \frac{6}{6}} + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.6.4

Combine the numerators over the common denominator.

$\frac{\frac{1}{q^{\frac{5}{6}}} - q^{\frac{- 5 + 6}{6}} + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.1.6.5

Add $- 5$ and $6$ .

$\frac{\frac{1}{q^{\frac{5}{6}}} - q^{\frac{1}{6}} + 6 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.3.2

Add $- q^{\frac{1}{6}}$ and $6 q^{\frac{1}{6}}$ .

$\frac{\frac{1}{q^{\frac{5}{6}}} + 5 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.4

Combine terms.

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Step 4.16.4.1

Multiply $\frac{\frac{1}{q^{\frac{5}{6}}} + 5 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$ by $\frac{q^{\frac{5}{6}}}{q^{\frac{5}{6}}}$ .

$\frac{q^{\frac{5}{6}}}{q^{\frac{5}{6}}} \cdot \frac{\frac{1}{q^{\frac{5}{6}}} + 5 q^{\frac{1}{6}}}{{(1 - q)}^{2}}$

Step 4.16.4.2

Combine.

$\frac{q^{\frac{5}{6}} (\frac{1}{q^{\frac{5}{6}}} + 5 q^{\frac{1}{6}})}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.3

Apply the distributive property.

$\frac{q^{\frac{5}{6}} \frac{1}{q^{\frac{5}{6}}} + q^{\frac{5}{6}} (5 q^{\frac{1}{6}})}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.4

Cancel the common factor of $q^{\frac{5}{6}}$ .

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Step 4.16.4.4.1

Cancel the common factor.

$\frac{q^{\frac{5}{6}} \frac{1}{q^{\frac{5}{6}}} + q^{\frac{5}{6}} (5 q^{\frac{1}{6}})}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.4.2

Rewrite the expression.

$\frac{1 + q^{\frac{5}{6}} (5 q^{\frac{1}{6}})}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.5

Multiply $q^{\frac{5}{6}}$ by $q^{\frac{1}{6}}$ by adding the exponents.

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Step 4.16.4.5.1

Move $q^{\frac{1}{6}}$ .

$\frac{1 + q^{\frac{1}{6}} q^{\frac{5}{6}} \cdot 5}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.5.2

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

$\frac{1 + q^{\frac{1}{6} + \frac{5}{6}} \cdot 5}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.5.3

Combine the numerators over the common denominator.

$\frac{1 + q^{\frac{1 + 5}{6}} \cdot 5}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.5.4

Add $1$ and $5$ .

$\frac{1 + q^{\frac{6}{6}} \cdot 5}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.5.5

Divide $6$ by $6$ .

$\frac{1 + q^{1} \cdot 5}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.6

Simplify $q^{1} \cdot 5$ .

$\frac{1 + q \cdot 5}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.4.7

Move $5$ to the left of $q$ .

$\frac{1 + 5 q}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 4.16.5

Reorder terms.

$\frac{5 q + 1}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 5

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

$p' = \frac{5 q + 1}{q^{\frac{5}{6}} {(1 - q)}^{2}}$

Step 6

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

$\frac{d p}{d q} = \frac{5 q + 1}{q^{\frac{5}{6}} {(1 - q)}^{2}}$