Calculus Examples

$f (x) = \frac{16}{3 {(x - 6)}^{\frac{1}{3}}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1.1

Since $\frac{16}{3}$ is constant with respect to $x$ , the derivative of $\frac{16}{3 {(x - 6)}^{\frac{1}{3}}}$ with respect to $x$ is $\frac{16}{3} \frac{d}{d x} [\frac{1}{{(x - 6)}^{\frac{1}{3}}}]$ .

$\frac{16}{3} \frac{d}{d x} [\frac{1}{{(x - 6)}^{\frac{1}{3}}}]$

Step 1.1.1.2

Apply basic rules of exponents.

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Step 1.1.1.2.1

Rewrite $\frac{1}{{(x - 6)}^{\frac{1}{3}}}$ as ${({(x - 6)}^{\frac{1}{3}})}^{- 1}$ .

$\frac{16}{3} \frac{d}{d x} [{({(x - 6)}^{\frac{1}{3}})}^{- 1}]$

Step 1.1.1.2.2

Multiply the exponents in ${({(x - 6)}^{\frac{1}{3}})}^{- 1}$ .

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Step 1.1.1.2.2.1

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

$\frac{16}{3} \frac{d}{d x} [{(x - 6)}^{\frac{1}{3} \cdot - 1}]$

Step 1.1.1.2.2.2

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{16}{3} \frac{d}{d x} [{(x - 6)}^{\frac{- 1}{3}}]$

Step 1.1.1.2.2.3

Move the negative in front of the fraction.

$\frac{16}{3} \frac{d}{d x} [{(x - 6)}^{- \frac{1}{3}}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{1}{3}}$ and $g (x) = x - 6$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x - 6$ .

$\frac{16}{3} (\frac{d}{d u} [u^{- \frac{1}{3}}] \frac{d}{d x} [x - 6])$

Step 1.1.2.2

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

$\frac{16}{3} (- \frac{1}{3} u^{- \frac{1}{3} - 1} \frac{d}{d x} [x - 6])$

Step 1.1.2.3

Replace all occurrences of $u$ with $x - 6$ .

$\frac{16}{3} (- \frac{1}{3} {(x - 6)}^{- \frac{1}{3} - 1} \frac{d}{d x} [x - 6])$

Step 1.1.3

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

$\frac{16}{3} (- \frac{1}{3} {(x - 6)}^{- \frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x - 6])$

Step 1.1.4

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{16}{3} (- \frac{1}{3} {(x - 6)}^{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x - 6])$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{16}{3} (- \frac{1}{3} {(x - 6)}^{\frac{- 1 - 1 \cdot 3}{3}} \frac{d}{d x} [x - 6])$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $3$ .

$\frac{16}{3} (- \frac{1}{3} {(x - 6)}^{\frac{- 1 - 3}{3}} \frac{d}{d x} [x - 6])$

Step 1.1.6.2

Subtract $3$ from $- 1$ .

$\frac{16}{3} (- \frac{1}{3} {(x - 6)}^{\frac{- 4}{3}} \frac{d}{d x} [x - 6])$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$\frac{16}{3} (- \frac{1}{3} {(x - 6)}^{- \frac{4}{3}} \frac{d}{d x} [x - 6])$

Step 1.1.7.2

Combine ${(x - 6)}^{- \frac{4}{3}}$ and $\frac{1}{3}$ .

$\frac{16}{3} (- \frac{{(x - 6)}^{- \frac{4}{3}}}{3} \frac{d}{d x} [x - 6])$

Step 1.1.7.3

Move ${(x - 6)}^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{16}{3} (- \frac{1}{3 {(x - 6)}^{\frac{4}{3}}} \frac{d}{d x} [x - 6])$

Step 1.1.7.4

Multiply $\frac{16}{3}$ by $\frac{1}{3 {(x - 6)}^{\frac{4}{3}}}$ .

$- \frac{16}{3 (3 {(x - 6)}^{\frac{4}{3}})} \frac{d}{d x} [x - 6]$

Step 1.1.7.5

Multiply $3$ by $3$ .

$- \frac{16}{9 {(x - 6)}^{\frac{4}{3}}} \frac{d}{d x} [x - 6]$

Step 1.1.8

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

$- \frac{16}{9 {(x - 6)}^{\frac{4}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 6])$

Step 1.1.9

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

$- \frac{16}{9 {(x - 6)}^{\frac{4}{3}}} (1 + \frac{d}{d x} [- 6])$

Step 1.1.10

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

$- \frac{16}{9 {(x - 6)}^{\frac{4}{3}}} (1 + 0)$

Step 1.1.11

Simplify the expression.

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Step 1.1.11.1

Add $1$ and $0$ .

$- \frac{16}{9 {(x - 6)}^{\frac{4}{3}}} \cdot 1$

Step 1.1.11.2

Multiply $- 1$ by $1$ .

$f' (x) = - \frac{16}{9 {(x - 6)}^{\frac{4}{3}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{16}{9 {(x - 6)}^{\frac{4}{3}}}$ .

$- \frac{16}{9 {(x - 6)}^{\frac{4}{3}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{16}{9 {(x - 6)}^{\frac{4}{3}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{16}{9 {(x - 6)}^{\frac{4}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$16 = 0$

Step 2.3

Since $16 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{16}{9 \sqrt[3]{{(x - 6)}^{4}}}$

Step 3.2

Set the denominator in $\frac{16}{9 \sqrt[3]{{(x - 6)}^{4}}}$ equal to $0$ to find where the expression is undefined.

$9 \sqrt[3]{{(x - 6)}^{4}} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${(9 \sqrt[3]{{(x - 6)}^{4}})}^{3} = 0^{3}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(x - 6)}^{4}}$ as ${(x - 6)}^{\frac{4}{3}}$ .

${(9 {(x - 6)}^{\frac{4}{3}})}^{3} = 0^{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(9 {(x - 6)}^{\frac{4}{3}})}^{3}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $9 {(x - 6)}^{\frac{4}{3}}$ .

$9^{3} {({(x - 6)}^{\frac{4}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.2

Raise $9$ to the power of $3$ .

$729 {({(x - 6)}^{\frac{4}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(x - 6)}^{\frac{4}{3}})}^{3}$ .

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Step 3.3.2.2.1.3.1

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

$729 {(x - 6)}^{\frac{4}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$729 {(x - 6)}^{\frac{4}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$729 {(x - 6)}^{4} = 0^{3}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$729 {(x - 6)}^{4} = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Divide each term in $729 {(x - 6)}^{4} = 0$ by $729$ and simplify.

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Step 3.3.3.1.1

Divide each term in $729 {(x - 6)}^{4} = 0$ by $729$ .

$\frac{729 {(x - 6)}^{4}}{729} = \frac{0}{729}$

Step 3.3.3.1.2

Simplify the left side.

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Step 3.3.3.1.2.1

Cancel the common factor of $729$ .

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Step 3.3.3.1.2.1.1

Cancel the common factor.

$\frac{729 {(x - 6)}^{4}}{729} = \frac{0}{729}$

Step 3.3.3.1.2.1.2

Divide ${(x - 6)}^{4}$ by $1$ .

${(x - 6)}^{4} = \frac{0}{729}$

Step 3.3.3.1.3

Simplify the right side.

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Step 3.3.3.1.3.1

Divide $0$ by $729$ .

${(x - 6)}^{4} = 0$

Step 3.3.3.2

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.3.3.3

Add $6$ to both sides of the equation.

$x = 6$

Step 4

Evaluate $\frac{16}{3 {(x - 6)}^{\frac{1}{3}}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 6$ .

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Step 4.1.1

Substitute $6$ for $x$ .

$\frac{16}{3 {((6) - 6)}^{\frac{1}{3}}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Subtract $6$ from $6$ .

$\frac{16}{3 \cdot 0^{\frac{1}{3}}}$

Step 4.1.2.1.2

Rewrite $0$ as $0^{3}$ .

$\frac{16}{3 \cdot {(0^{3})}^{\frac{1}{3}}}$

Step 4.1.2.1.3

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

$\frac{16}{3 \cdot 0^{3 (\frac{1}{3})}}$

Step 4.1.2.2

Cancel the common factor of $3$ .

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Step 4.1.2.2.1

Cancel the common factor.

$\frac{16}{3 \cdot 0^{3 (\frac{1}{3})}}$

Step 4.1.2.2.2

Rewrite the expression.

$\frac{16}{3 \cdot 0^{1}}$

Step 4.1.2.3

Simplify the expression.

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Step 4.1.2.3.1

Evaluate the exponent.

$\frac{16}{3 \cdot 0}$

Step 4.1.2.3.2

Multiply $3$ by $0$ .

$\frac{16}{0}$

Step 4.1.2.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{16}{0}$

Step 4.1.2.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found