Calculus Examples

$f (x) = x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}$

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 Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = {(x + 3)}^{\frac{2}{3}}$ .

$f' (x) = x^{\frac{1}{3}} \frac{d}{d x} ({(x + 3)}^{\frac{2}{3}}) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

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 $f (x) = x^{\frac{2}{3}}$ and $g (x) = x + 3$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x + 3$ .

$f' (x) = x^{\frac{1}{3}} (\frac{d}{d u} (u^{\frac{2}{3}}) \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

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 $n = \frac{2}{3}$ .

$f' (x) = x^{\frac{1}{3}} (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.2.3

Replace all occurrences of $u$ with $x + 3$ .

$f' (x) = x^{\frac{1}{3}} (\frac{2}{3} \cdot {(x + 3)}^{\frac{2}{3} - 1} \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.3

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

$f' (x) = x^{\frac{1}{3}} (\frac{2}{3} \cdot {(x + 3)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.4

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

$f' (x) = x^{\frac{1}{3}} (\frac{2}{3} \cdot {(x + 3)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.5

Combine the numerators over the common denominator.

$f' (x) = x^{\frac{1}{3}} (\frac{2}{3} \cdot {(x + 3)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $3$ .

$f' (x) = x^{\frac{1}{3}} (\frac{2}{3} \cdot {(x + 3)}^{\frac{2 - 3}{3}} \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.6.2

Subtract $3$ from $2$ .

$f' (x) = x^{\frac{1}{3}} (\frac{2}{3} \cdot {(x + 3)}^{\frac{- 1}{3}} \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$f' (x) = x^{\frac{1}{3}} (\frac{2}{3} \cdot {(x + 3)}^{- \frac{1}{3}} \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.7.2

Combine $\frac{2}{3}$ and ${(x + 3)}^{- \frac{1}{3}}$ .

$f' (x) = x^{\frac{1}{3}} (\frac{2 {(x + 3)}^{- \frac{1}{3}}}{3} \cdot \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.7.3

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

$f' (x) = x^{\frac{1}{3}} (\frac{2}{3 {(x + 3)}^{\frac{1}{3}}} \cdot \frac{d}{d x} (x + 3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.7.4

Combine $\frac{2}{3 {(x + 3)}^{\frac{1}{3}}}$ and $x^{\frac{1}{3}}$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} \cdot \frac{d}{d x} (x + 3) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.8

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} \cdot (\frac{d}{d x} (x) + \frac{d}{d x} (3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.9

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

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} \cdot (1 + \frac{d}{d x} (3)) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.10

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} \cdot (1 + 0) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.11

Simplify the expression.

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Step 1.1.11.1

Add $1$ and $0$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} \cdot 1 + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.11.2

Multiply $\frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}}$ by $1$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} (x^{\frac{1}{3}})$

Step 1.1.12

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

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + {(x + 3)}^{\frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1})$

Step 1.1.13

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

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + {(x + 3)}^{\frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 1.1.14

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

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + {(x + 3)}^{\frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.1.15

Combine the numerators over the common denominator.

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + {(x + 3)}^{\frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 1.1.16

Simplify the numerator.

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Step 1.1.16.1

Multiply $- 1$ by $3$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + {(x + 3)}^{\frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Step 1.1.16.2

Subtract $3$ from $1$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + {(x + 3)}^{\frac{2}{3}} (\frac{1}{3} x^{\frac{- 2}{3}})$

Step 1.1.17

Move the negative in front of the fraction.

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + {(x + 3)}^{\frac{2}{3}} (\frac{1}{3} x^{- \frac{2}{3}})$

Step 1.1.18

Combine $\frac{1}{3}$ and $x^{- \frac{2}{3}}$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + {(x + 3)}^{\frac{2}{3}} (\frac{x^{- \frac{2}{3}}}{3})$

Step 1.1.19

Combine ${(x + 3)}^{\frac{2}{3}}$ and $\frac{x^{- \frac{2}{3}}}{3}$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + \frac{{(x + 3)}^{\frac{2}{3}} x^{- \frac{2}{3}}}{3}$

Step 1.1.20

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} + \frac{{(x + 3)}^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$

Step 1.1.21

To write $\frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} \cdot \frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}} + \frac{{(x + 3)}^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$

Step 1.1.22

To write $\frac{{(x + 3)}^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{{(x + 3)}^{\frac{1}{3}}}{{(x + 3)}^{\frac{1}{3}}}$ .

$f' (x) = \frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} \cdot \frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}} + \frac{{(x + 3)}^{\frac{2}{3}}}{3 x^{\frac{2}{3}}} \cdot \frac{{(x + 3)}^{\frac{1}{3}}}{{(x + 3)}^{\frac{1}{3}}}$

Step 1.1.23

Write each expression with a common denominator of $3 {(x + 3)}^{\frac{1}{3}} x^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.23.1

Multiply $\frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}}$ by $\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ .

$f' (x) = \frac{2 x^{\frac{1}{3}} x^{\frac{2}{3}}}{3 {(x + 3)}^{\frac{1}{3}} x^{\frac{2}{3}}} + \frac{{(x + 3)}^{\frac{2}{3}}}{3 x^{\frac{2}{3}}} \cdot \frac{{(x + 3)}^{\frac{1}{3}}}{{(x + 3)}^{\frac{1}{3}}}$

Step 1.1.23.2

Multiply $\frac{{(x + 3)}^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}$ by $\frac{{(x + 3)}^{\frac{1}{3}}}{{(x + 3)}^{\frac{1}{3}}}$ .

$f' (x) = \frac{2 x^{\frac{1}{3}} x^{\frac{2}{3}}}{3 {(x + 3)}^{\frac{1}{3}} x^{\frac{2}{3}}} + \frac{{(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.23.3

Reorder the factors of $3 {(x + 3)}^{\frac{1}{3}} x^{\frac{2}{3}}$ .

$f' (x) = \frac{2 x^{\frac{1}{3}} x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}} + \frac{{(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.24

Combine the numerators over the common denominator.

$f' (x) = \frac{2 x^{\frac{1}{3}} x^{\frac{2}{3}} + {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.25

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 1.1.25.1

Move $x^{\frac{2}{3}}$ .

$f' (x) = \frac{2 (x^{\frac{2}{3}} x^{\frac{1}{3}}) + {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.25.2

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

$f' (x) = \frac{2 x^{\frac{2}{3} + \frac{1}{3}} + {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.25.3

Combine the numerators over the common denominator.

$f' (x) = \frac{2 x^{\frac{2 + 1}{3}} + {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.25.4

Add $2$ and $1$ .

$f' (x) = \frac{2 x^{\frac{3}{3}} + {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.25.5

Divide $3$ by $3$ .

$f' (x) = \frac{2 x + {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.26

Simplify $2 x^{1}$ .

$f' (x) = \frac{2 x + {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.27

Multiply ${(x + 3)}^{\frac{2}{3}}$ by ${(x + 3)}^{\frac{1}{3}}$ by adding the exponents.

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Step 1.1.27.1

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

$f' (x) = \frac{2 x + {(x + 3)}^{\frac{2}{3} + \frac{1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.27.2

Combine the numerators over the common denominator.

$f' (x) = \frac{2 x + {(x + 3)}^{\frac{2 + 1}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.27.3

Add $2$ and $1$ .

$f' (x) = \frac{2 x + {(x + 3)}^{\frac{3}{3}}}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.27.4

Divide $3$ by $3$ .

$f' (x) = \frac{2 x + x + 3}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.28

Simplify ${(x + 3)}^{1}$ .

$f' (x) = \frac{2 x + x + 3}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.29

Add $2 x$ and $x$ .

$f' (x) = \frac{3 x + 3}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.30

Factor $3$ out of $3 x$ .

$f' (x) = \frac{3 (x) + 3}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.31

Factor $3$ out of $3$ .

$f' (x) = \frac{3 (x) + 3 \cdot 1}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.32

Factor $3$ out of $3 (x) + 3 (1)$ .

$f' (x) = \frac{3 (x + 1)}{3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.1.33

Cancel the common factors.

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Step 1.1.33.1

Factor $3$ out of $3 x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}$ .

$f' (x) = \frac{3 (x + 1)}{3 (x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}})}$

Step 1.1.33.2

Cancel the common factor.

$f' (x) = \frac{3 (x + 1)}{3 (x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}})}$

Step 1.1.33.3

Rewrite the expression.

$f' (x) = \frac{x + 1}{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x + 1}{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$ .

$\frac{x + 1}{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x + 1}{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x + 1}{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$x + 1 = 0$

Step 2.3

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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

$\frac{x + 1}{\sqrt[3]{x^{2}} {(x + 3)}^{\frac{1}{3}}}$

Step 3.1.2

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

$\frac{x + 1}{\sqrt[3]{x^{2}} \sqrt[3]{{(x + 3)}^{1}}}$

Step 3.1.3

Anything raised to $1$ is the base itself.

$\frac{x + 1}{\sqrt[3]{x^{2}} \sqrt[3]{x + 3}}$

Step 3.2

Set the denominator in $\frac{x + 1}{\sqrt[3]{x^{2}} \sqrt[3]{x + 3}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x^{2}} \sqrt[3]{x + 3} = 0$

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.

${(\sqrt[3]{x^{2}} \sqrt[3]{x + 3})}^{3} = 0^{3}$

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 $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

${(x^{\frac{2}{3}} \sqrt[3]{x + 3})}^{3} = 0^{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(x^{\frac{2}{3}} \sqrt[3]{x + 3})}^{3}$ .

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Step 3.3.2.2.1.1

Apply basic rules of exponents.

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Step 3.3.2.2.1.1.1

Apply the product rule to $x^{\frac{2}{3}} \sqrt[3]{x + 3}$ .

${(x^{\frac{2}{3}})}^{3} {\sqrt[3]{x + 3}}^{3} = 0^{3}$

Step 3.3.2.2.1.1.2

Multiply the exponents in ${(x^{\frac{2}{3}})}^{3}$ .

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Step 3.3.2.2.1.1.2.1

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

$x^{\frac{2}{3} \cdot 3} {\sqrt[3]{x + 3}}^{3} = 0^{3}$

Step 3.3.2.2.1.1.2.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.1.2.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot 3} {\sqrt[3]{x + 3}}^{3} = 0^{3}$

Step 3.3.2.2.1.1.2.2.2

Rewrite the expression.

$x^{2} {\sqrt[3]{x + 3}}^{3} = 0^{3}$

Step 3.3.2.2.1.2

Rewrite ${\sqrt[3]{x + 3}}^{3}$ as $x + 3$ .

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Step 3.3.2.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 3}$ as ${(x + 3)}^{\frac{1}{3}}$ .

$x^{2} {({(x + 3)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.2.2

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

$x^{2} {(x + 3)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.2.3

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

$x^{2} {(x + 3)}^{\frac{3}{3}} = 0^{3}$

Step 3.3.2.2.1.2.4

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.2.4.1

Cancel the common factor.

$x^{2} {(x + 3)}^{\frac{3}{3}} = 0^{3}$

Step 3.3.2.2.1.2.4.2

Rewrite the expression.

$x^{2} {(x + 3)}^{1} = 0^{3}$

Step 3.3.2.2.1.2.5

Simplify.

$x^{2} (x + 3) = 0^{3}$

Step 3.3.2.2.1.3

Apply the distributive property.

$x^{2} x + x^{2} \cdot 3 = 0^{3}$

Step 3.3.2.2.1.4

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

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Step 3.3.2.2.1.4.1

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

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Step 3.3.2.2.1.4.1.1

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

$x^{2} x^{1} + x^{2} \cdot 3 = 0^{3}$

Step 3.3.2.2.1.4.1.2

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

$x^{2 + 1} + x^{2} \cdot 3 = 0^{3}$

Step 3.3.2.2.1.4.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot 3 = 0^{3}$

Step 3.3.2.2.1.5

Move $3$ to the left of $x^{2}$ .

$x^{3} + 3 x^{2} = 0^{3}$

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 $0$ .

$x^{3} + 3 x^{2} = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Factor $x^{2}$ out of $x^{3} + 3 x^{2}$ .

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Step 3.3.3.1.1

Factor $x^{2}$ out of $x^{3}$ .

$x^{2} x + 3 x^{2} = 0$

Step 3.3.3.1.2

Factor $x^{2}$ out of $3 x^{2}$ .

$x^{2} x + x^{2} \cdot 3 = 0$

Step 3.3.3.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 3$ .

$x^{2} (x + 3) = 0$

Step 3.3.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x + 3 = 0$

Step 3.3.3.3

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 3.3.3.3.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 3.3.3.3.2

Solve $x^{2} = 0$ for $x$ .

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Step 3.3.3.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 3.3.3.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.3.3.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.3.3.3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.3.3.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.3.3.4

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 3.3.3.4.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.3.3.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.3.3.5

The final solution is all the values that make $x^{2} (x + 3) = 0$ true.

$x = 0, - 3$

Step 3.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 3, x = 0$

Step 4

Evaluate $x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 1$ .

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Step 4.1.1

Substitute $- 1$ for $x$ .

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

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

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 4.1.2.1.2

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

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

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.

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

Step 4.1.2.2.2

Rewrite the expression.

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

Step 4.1.2.3

Evaluate the exponent.

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

Step 4.1.2.4

Add $- 1$ and $3$ .

$- 1 \cdot 2^{\frac{2}{3}}$

Step 4.1.2.5

Rewrite $- 1 \cdot 2^{\frac{2}{3}}$ as $- 2^{\frac{2}{3}}$ .

$- 2^{\frac{2}{3}}$

Step 4.2

Evaluate at $x = - 3$ .

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Step 4.2.1

Substitute $- 3$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the expression.

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Step 4.2.2.1.1

Add $- 3$ and $3$ .

${(- 3)}^{\frac{1}{3}} \cdot 0^{\frac{2}{3}}$

Step 4.2.2.1.2

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

${(- 3)}^{\frac{1}{3}} \cdot {(0^{3})}^{\frac{2}{3}}$

Step 4.2.2.1.3

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

${(- 3)}^{\frac{1}{3}} \cdot 0^{3 (\frac{2}{3})}$

Step 4.2.2.2

Cancel the common factor of $3$ .

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Step 4.2.2.2.1

Cancel the common factor.

${(- 3)}^{\frac{1}{3}} \cdot 0^{3 (\frac{2}{3})}$

Step 4.2.2.2.2

Rewrite the expression.

${(- 3)}^{\frac{1}{3}} \cdot 0^{2}$

Step 4.2.2.3

Simplify the expression.

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Step 4.2.2.3.1

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

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

Step 4.2.2.3.2

Multiply ${(- 3)}^{\frac{1}{3}}$ by $0$ .

$0$

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

${(0)}^{\frac{1}{3}} {((0) + 3)}^{\frac{2}{3}}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the expression.

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Step 4.3.2.1.1

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

${(0^{3})}^{\frac{1}{3}} {((0) + 3)}^{\frac{2}{3}}$

Step 4.3.2.1.2

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

$0^{3 (\frac{1}{3})} {((0) + 3)}^{\frac{2}{3}}$

Step 4.3.2.2

Cancel the common factor of $3$ .

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Step 4.3.2.2.1

Cancel the common factor.

$0^{3 (\frac{1}{3})} {((0) + 3)}^{\frac{2}{3}}$

Step 4.3.2.2.2

Rewrite the expression.

$0^{1} {((0) + 3)}^{\frac{2}{3}}$

Step 4.3.2.3

Evaluate the exponent.

$0 {((0) + 3)}^{\frac{2}{3}}$

Step 4.3.2.4

Add $0$ and $3$ .

$0 \cdot 3^{\frac{2}{3}}$

Step 4.3.2.5

Multiply $0$ by $3^{\frac{2}{3}}$ .

$0$

Step 4.4

List all of the points.

$(- 1, - 2^{\frac{2}{3}}), (- 3, 0), (0, 0)$

Step 5