Calculus Examples

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

Step 1

Find the first derivative of the function.

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

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

Step 1.2

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 1.2.1

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

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

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

$x^{\frac{1}{3}} (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.2.3

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

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

Step 1.3

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

$x^{\frac{1}{3}} (\frac{2}{3} {(x + 3)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.4

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

$x^{\frac{1}{3}} (\frac{2}{3} {(x + 3)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.5

Combine the numerators over the common denominator.

$x^{\frac{1}{3}} (\frac{2}{3} {(x + 3)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [x + 3]) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $3$ .

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

Step 1.6.2

Subtract $3$ from $2$ .

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

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.7.4

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

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

$\frac{2 x^{\frac{1}{3}}}{3 {(x + 3)}^{\frac{1}{3}}} (1 + 0) + {(x + 3)}^{\frac{2}{3}} \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.11

Simplify the expression.

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Step 1.11.1

Add $1$ and $0$ .

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

Step 1.11.2

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

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

Step 1.12

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

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

Step 1.13

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

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

Step 1.14

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

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

Step 1.15

Combine the numerators over the common denominator.

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

Step 1.16

Simplify the numerator.

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Step 1.16.1

Multiply $- 1$ by $3$ .

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

Step 1.16.2

Subtract $3$ from $1$ .

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

Step 1.17

Move the negative in front of the fraction.

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

Step 1.18

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

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

Step 1.19

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

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

Step 1.20

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

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

Step 1.21

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

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

Step 1.22

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

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

Step 1.23

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 1.23.1

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

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

Step 1.23.2

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

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

Step 1.23.3

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

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

Step 1.24

Combine the numerators over the common denominator.

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

Step 1.25

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

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Step 1.25.1

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

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

Step 1.25.2

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

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

Step 1.25.3

Combine the numerators over the common denominator.

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

Step 1.25.4

Add $2$ and $1$ .

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

Step 1.25.5

Divide $3$ by $3$ .

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

Step 1.26

Simplify $2 x^{1}$ .

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

Step 1.27

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

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Step 1.27.1

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

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

Step 1.27.2

Combine the numerators over the common denominator.

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

Step 1.27.3

Add $2$ and $1$ .

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

Step 1.27.4

Divide $3$ by $3$ .

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

Step 1.28

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

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

Step 1.29

Add $2 x$ and $x$ .

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

Step 1.30

Factor $3$ out of $3 x$ .

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

Step 1.31

Factor $3$ out of $3$ .

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

Step 1.32

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

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

Step 1.33

Cancel the common factors.

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Step 1.33.1

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

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

Step 1.33.2

Cancel the common factor.

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

Step 1.33.3

Rewrite the expression.

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add $1$ and $0$ .

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

Step 2.2.4.2

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

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

Step 2.3

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

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

Step 2.4

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 $g (x) = x + 3$ .

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $3$ .

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

Step 2.8.2

Subtract $3$ from $1$ .

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

Step 2.9

Combine fractions.

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Step 2.9.1

Move the negative in front of the fraction.

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

Step 2.9.2

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

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

Step 2.9.3

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

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

Step 2.9.4

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

Simplify the expression.

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Step 2.13.1

Add $1$ and $0$ .

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

Step 2.13.2

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

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

Step 2.14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Combine the numerators over the common denominator.

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

Step 2.18

Simplify the numerator.

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Step 2.18.1

Multiply $- 1$ by $3$ .

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

Step 2.18.2

Subtract $3$ from $2$ .

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

Step 2.19

Move the negative in front of the fraction.

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

Step 2.20

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

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

Step 2.21

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

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

Step 2.22

Simplify the expression.

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Step 2.22.1

Move $2$ to the left of ${(x + 3)}^{\frac{1}{3}}$ .

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

Step 2.22.2

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

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

Step 2.23

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

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

Step 2.24

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

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

Step 2.25

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

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Step 2.25.1

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

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

Step 2.25.2

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

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

Step 2.25.3

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

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

Step 2.26

Combine the numerators over the common denominator.

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

Step 2.27

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

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Step 2.27.1

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

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

Step 2.27.2

Combine the numerators over the common denominator.

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

Step 2.27.3

Add $2$ and $1$ .

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

Step 2.27.4

Divide $3$ by $3$ .

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

Step 2.28

Simplify $x^{1}$ .

Step 2.29

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

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Step 2.29.1

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

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

Step 2.29.2

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

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

Step 2.29.3

Combine the numerators over the common denominator.

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

Step 2.29.4

Add $2$ and $1$ .

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

Step 2.29.5

Divide $3$ by $3$ .

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

Step 2.30

Simplify $2 {(x + 3)}^{1}$ .

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

Step 2.31

Simplify.

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Step 2.31.1

Apply the product rule to $x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}$ .

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

Step 2.31.2

Apply the distributive property.

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

Step 2.31.3

Apply the distributive property.

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

Step 2.31.4

Simplify the numerator.

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Step 2.31.4.1

Simplify the numerator.

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Step 2.31.4.1.1

Multiply $2$ by $3$ .

$f'' (x) = \frac{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}} + (- x - 1 \cdot 1) (\frac{x + 2 x + 6}{3 x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}})}{{(x^{\frac{2}{3}})}^{2} {({(x + 3)}^{\frac{1}{3}})}^{2}}$

Step 2.31.4.1.2

Add $x$ and $2 x$ .

$f'' (x) = \frac{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}} + (- x - 1 \cdot 1) (\frac{3 x + 6}{3 x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}})}{{(x^{\frac{2}{3}})}^{2} {({(x + 3)}^{\frac{1}{3}})}^{2}}$

Step 2.31.4.1.3

Factor $3$ out of $3 x + 6$ .

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Step 2.31.4.1.3.1

Factor $3$ out of $3 x$ .

$f'' (x) = \frac{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}} + (- x - 1 \cdot 1) (\frac{3 (x) + 6}{3 x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}})}{{(x^{\frac{2}{3}})}^{2} {({(x + 3)}^{\frac{1}{3}})}^{2}}$

Step 2.31.4.1.3.2

Factor $3$ out of $6$ .

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

Step 2.31.4.1.3.3

Factor $3$ out of $3 x + 3 \cdot 2$ .

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

Step 2.31.4.2

Multiply $- 1$ by $1$ .

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

Step 2.31.4.3

Cancel the common factor.

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

Step 2.31.4.4

Rewrite the expression.

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

Step 2.31.4.5

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

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

Step 2.31.4.6

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

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

Step 2.31.4.7

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

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

Step 2.31.4.8

Combine the numerators over the common denominator.

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

Step 2.31.4.9

Rewrite $\frac{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}} (x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}) + (- x - 1) (x + 2)}{x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}}$ in a factored form.

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Step 2.31.4.9.1

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

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Step 2.31.4.9.1.1

Move $x^{\frac{1}{3}}$ .

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

Step 2.31.4.9.1.2

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

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

Step 2.31.4.9.1.3

Combine the numerators over the common denominator.

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

Step 2.31.4.9.1.4

Add $1$ and $2$ .

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

Step 2.31.4.9.1.5

Divide $3$ by $3$ .

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

Step 2.31.4.9.2

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

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

Step 2.31.4.9.3

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

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Step 2.31.4.9.3.1

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

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

Step 2.31.4.9.3.2

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

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

Step 2.31.4.9.3.3

Combine the numerators over the common denominator.

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

Step 2.31.4.9.3.4

Add $2$ and $1$ .

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

Step 2.31.4.9.3.5

Divide $3$ by $3$ .

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

Step 2.31.4.9.4

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

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

Step 2.31.4.9.5

Apply the distributive property.

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

Step 2.31.4.9.6

Multiply $x$ by $x$ .

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

Step 2.31.4.9.7

Move $3$ to the left of $x$ .

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

Step 2.31.4.9.8

Expand $(- x - 1) (x + 2)$ using the FOIL Method.

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Step 2.31.4.9.8.1

Apply the distributive property.

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

Step 2.31.4.9.8.2

Apply the distributive property.

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

Step 2.31.4.9.8.3

Apply the distributive property.

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

Step 2.31.4.9.9

Simplify and combine like terms.

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Step 2.31.4.9.9.1

Simplify each term.

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Step 2.31.4.9.9.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.31.4.9.9.1.1.1

Move $x$ .

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

Step 2.31.4.9.9.1.1.2

Multiply $x$ by $x$ .

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

Step 2.31.4.9.9.1.2

Multiply $2$ by $- 1$ .

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

Step 2.31.4.9.9.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.31.4.9.9.1.4

Multiply $- 1$ by $2$ .

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

Step 2.31.4.9.9.2

Subtract $x$ from $- 2 x$ .

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

Step 2.31.4.9.10

Subtract $x^{2}$ from $x^{2}$ .

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

Step 2.31.4.9.11

Add $3 x$ and $0$ .

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

Step 2.31.4.9.12

Subtract $3 x$ from $3 x$ .

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

Step 2.31.4.9.13

Subtract $2$ from $0$ .

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

Step 2.31.4.10

Move the negative in front of the fraction.

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

Step 2.31.5

Combine terms.

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Step 2.31.5.1

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

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Step 2.31.5.1.1

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

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

Step 2.31.5.1.2

Multiply $\frac{2}{3} \cdot 2$ .

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Step 2.31.5.1.2.1

Combine $\frac{2}{3}$ and $2$ .

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

Step 2.31.5.1.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{- \frac{2}{x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}}}{x^{\frac{4}{3}} {({(x + 3)}^{\frac{1}{3}})}^{2}}$

Step 2.31.5.2

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

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Step 2.31.5.2.1

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

$f'' (x) = \frac{- \frac{2}{x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}}}{x^{\frac{4}{3}} {(x + 3)}^{\frac{1}{3} \cdot 2}}$

Step 2.31.5.2.2

Combine $\frac{1}{3}$ and $2$ .

$f'' (x) = \frac{- \frac{2}{x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}}}{x^{\frac{4}{3}} {(x + 3)}^{\frac{2}{3}}}$

Step 2.31.5.3

Rewrite $\frac{- \frac{2}{x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}}}{x^{\frac{4}{3}} {(x + 3)}^{\frac{2}{3}}}$ as a product.

$f'' (x) = - \frac{2}{x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}} \cdot \frac{1}{x^{\frac{4}{3}} {(x + 3)}^{\frac{2}{3}}}$

Step 2.31.5.4

Multiply $\frac{1}{x^{\frac{4}{3}} {(x + 3)}^{\frac{2}{3}}}$ by $\frac{2}{x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}}$ .

$f'' (x) = - \frac{2}{x^{\frac{4}{3}} {(x + 3)}^{\frac{2}{3}} (x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}})}$

Step 2.31.5.5

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

$f'' (x) = - \frac{2}{x^{\frac{1}{3} + \frac{4}{3}} {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{2}{3}}}$

Step 2.31.5.6

Combine the numerators over the common denominator.

$f'' (x) = - \frac{2}{x^{\frac{1 + 4}{3}} {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{2}{3}}}$

Step 2.31.5.7

Add $1$ and $4$ .

$f'' (x) = - \frac{2}{x^{\frac{5}{3}} {(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{2}{3}}}$

Step 2.31.5.8

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

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Step 2.31.5.8.1

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

$f'' (x) = - \frac{2}{x^{\frac{5}{3}} ({(x + 3)}^{\frac{2}{3}} {(x + 3)}^{\frac{2}{3}})}$

Step 2.31.5.8.2

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

$f'' (x) = - \frac{2}{x^{\frac{5}{3}} {(x + 3)}^{\frac{2}{3} + \frac{2}{3}}}$

Step 2.31.5.8.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{2}{x^{\frac{5}{3}} {(x + 3)}^{\frac{2 + 2}{3}}}$

Step 2.31.5.8.4

Add $2$ and $2$ .

$f'' (x) = - \frac{2}{x^{\frac{5}{3}} {(x + 3)}^{\frac{4}{3}}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

Step 4.1.2

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 4.1.2.1

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 4.1.2.2

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 4.1.2.3

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 4.1.3

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 4.1.4

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 4.1.5

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 4.1.6

Simplify the numerator.

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Step 4.1.6.1

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 4.1.6.2

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 4.1.7

Combine fractions.

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Step 4.1.7.1

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 4.1.7.2

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 4.1.7.3

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 4.1.7.4

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 4.1.8

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 4.1.9

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 4.1.10

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 4.1.11

Simplify the expression.

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Step 4.1.11.1

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 4.1.11.2

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 4.1.12

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 4.1.13

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 4.1.14

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 4.1.15

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 4.1.16

Simplify the numerator.

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Step 4.1.16.1

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 4.1.16.2

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 4.1.17

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 4.1.18

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 4.1.19

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 4.1.20

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 4.1.21

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 4.1.22

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 4.1.23

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 4.1.23.1

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 4.1.23.2

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 4.1.23.3

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 4.1.24

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 4.1.25

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

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Step 4.1.25.1

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 4.1.25.2

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 4.1.25.3

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 4.1.25.4

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 4.1.25.5

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 4.1.26

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 4.1.27

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

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Step 4.1.27.1

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 4.1.27.2

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 4.1.27.3

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 4.1.27.4

Divide $3$ by $3$ .

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

Step 4.1.28

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

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

Step 4.1.29

Add $2 x$ and $x$ .

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

Step 4.1.30

Factor $3$ out of $3 x$ .

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

Step 4.1.31

Factor $3$ out of $3$ .

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

Step 4.1.32

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 4.1.33

Cancel the common factors.

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Step 4.1.33.1

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 4.1.33.2

Cancel the common factor.

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

Step 4.1.33.3

Rewrite the expression.

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

Step 4.2

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 5

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 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$x + 1 = 0$

Step 5.3

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

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 6.1.2

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 6.1.3

Anything raised to $1$ is the base itself.

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

Step 6.2

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 6.3

Solve for $x$ .

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Step 6.3.1

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 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

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 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

Apply basic rules of exponents.

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Step 6.3.2.2.1.1.1

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 6.3.2.2.1.1.2

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

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Step 6.3.2.2.1.1.2.1

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 6.3.2.2.1.1.2.2

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.1.2.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.1.2.2.2

Rewrite the expression.

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

Step 6.3.2.2.1.2

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

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Step 6.3.2.2.1.2.1

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 6.3.2.2.1.2.2

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 6.3.2.2.1.2.3

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

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

Step 6.3.2.2.1.2.4

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.2.4.1

Cancel the common factor.

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

Step 6.3.2.2.1.2.4.2

Rewrite the expression.

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

Step 6.3.2.2.1.2.5

Simplify.

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

Step 6.3.2.2.1.3

Apply the distributive property.

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

Step 6.3.2.2.1.4

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

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Step 6.3.2.2.1.4.1

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

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Step 6.3.2.2.1.4.1.1

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

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

Step 6.3.2.2.1.4.1.2

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

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

Step 6.3.2.2.1.4.2

Add $2$ and $1$ .

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

Step 6.3.2.2.1.5

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

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

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

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

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

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

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

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Step 6.3.3.1.1

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

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

Step 6.3.3.1.2

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

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

Step 6.3.3.1.3

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

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

Step 6.3.3.2

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 6.3.3.3

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

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Step 6.3.3.3.1

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

$x^{2} = 0$

Step 6.3.3.3.2

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

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Step 6.3.3.3.2.1

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

$x = \pm \sqrt{0}$

Step 6.3.3.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.3.3.3.2.2.1

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

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

Step 6.3.3.3.2.2.2

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

$x = \pm 0$

Step 6.3.3.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.3.3.4

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

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Step 6.3.3.4.1

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

$x + 3 = 0$

Step 6.3.3.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6.3.3.5

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

$x = 0, - 3$

Step 6.4

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 7

Critical points to evaluate.

$x = - 1, - 3, 0$

Step 8

Evaluate the second derivative at $x = - 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2}{{(- 1)}^{\frac{5}{3}} {((- 1) + 3)}^{\frac{4}{3}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

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

$- \frac{2}{{({(- 1)}^{3})}^{\frac{5}{3}} {(- 1 + 3)}^{\frac{4}{3}}}$

Step 9.1.2

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

$- \frac{2}{{(- 1)}^{3 (\frac{5}{3})} {(- 1 + 3)}^{\frac{4}{3}}}$

Step 9.1.3

Cancel the common factor of $3$ .

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Step 9.1.3.1

Cancel the common factor.

$- \frac{2}{{(- 1)}^{3 (\frac{5}{3})} {(- 1 + 3)}^{\frac{4}{3}}}$

Step 9.1.3.2

Rewrite the expression.

$- \frac{2}{{(- 1)}^{5} {(- 1 + 3)}^{\frac{4}{3}}}$

Step 9.1.4

Raise $- 1$ to the power of $5$ .

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

Step 9.1.5

Add $- 1$ and $3$ .

$- \frac{2}{- 1 \cdot 2^{\frac{4}{3}}}$

Step 9.2

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

$- \frac{1}{- 1 \cdot 2^{\frac{4}{3}} \cdot 2^{- 1}}$

Step 9.3

Multiply $2^{\frac{4}{3}}$ by $2^{- 1}$ by adding the exponents.

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Step 9.3.1

Move $2^{- 1}$ .

$- \frac{1}{- 1 \cdot (2^{- 1} \cdot 2^{\frac{4}{3}})}$

Step 9.3.2

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

$- \frac{1}{- 1 \cdot 2^{- 1 + \frac{4}{3}}}$

Step 9.3.3

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

$- \frac{1}{- 1 \cdot 2^{- 1 \cdot \frac{3}{3} + \frac{4}{3}}}$

Step 9.3.4

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

$- \frac{1}{- 1 \cdot 2^{\frac{- 1 \cdot 3}{3} + \frac{4}{3}}}$

Step 9.3.5

Combine the numerators over the common denominator.

$- \frac{1}{- 1 \cdot 2^{\frac{- 1 \cdot 3 + 4}{3}}}$

Step 9.3.6

Simplify the numerator.

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Step 9.3.6.1

Multiply $- 1$ by $3$ .

$- \frac{1}{- 1 \cdot 2^{\frac{- 3 + 4}{3}}}$

Step 9.3.6.2

Add $- 3$ and $4$ .

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

Step 9.4

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

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

Step 9.5

Move the negative in front of the fraction.

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

Step 9.6

Multiply $- - \frac{1}{2^{\frac{1}{3}}}$ .

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Step 9.6.1

Multiply $- 1$ by $- 1$ .

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

Step 9.6.2

Multiply $\frac{1}{2^{\frac{1}{3}}}$ by $1$ .

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

Step 10

$x = - 1$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 1$ is a local minimum

Step 11

Find the y-value when $x = - 1$ .

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Step 11.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify the expression.

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Step 11.2.1.1

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

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

Step 11.2.1.2

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

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

Step 11.2.2

Cancel the common factor of $3$ .

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Step 11.2.2.1

Cancel the common factor.

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

Step 11.2.2.2

Rewrite the expression.

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

Step 11.2.3

Evaluate the exponent.

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

Step 11.2.4

Add $- 1$ and $3$ .

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

Step 11.2.5

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

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

Step 11.2.6

The final answer is $- 2^{\frac{2}{3}}$ .

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

Step 12

Evaluate the second derivative at $x = - 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2}{{(- 3)}^{\frac{5}{3}} {((- 3) + 3)}^{\frac{4}{3}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify the expression.

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Step 13.1.1

Add $- 3$ and $3$ .

$- \frac{2}{{(- 3)}^{\frac{5}{3}} \cdot 0^{\frac{4}{3}}}$

Step 13.1.2

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

$- \frac{2}{{(- 3)}^{\frac{5}{3}} \cdot {(0^{3})}^{\frac{4}{3}}}$

Step 13.1.3

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

$- \frac{2}{{(- 3)}^{\frac{5}{3}} \cdot 0^{3 (\frac{4}{3})}}$

Step 13.2

Cancel the common factor of $3$ .

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Step 13.2.1

Cancel the common factor.

$- \frac{2}{{(- 3)}^{\frac{5}{3}} \cdot 0^{3 (\frac{4}{3})}}$

Step 13.2.2

Rewrite the expression.

$- \frac{2}{{(- 3)}^{\frac{5}{3}} \cdot 0^{4}}$

Step 13.3

Simplify the expression.

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Step 13.3.1

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

$- \frac{2}{{(- 3)}^{\frac{5}{3}} \cdot 0}$

Step 13.3.2

Multiply ${(- 3)}^{\frac{5}{3}}$ by $0$ .

$- \frac{2}{0}$

Step 13.3.3

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

Undefined

$- \frac{2}{0}$

Step 13.4

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

Undefined

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 14.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - 3) \cup (- 3, - 1) \cup (- 1, 0) \cup (0, \infty)$

Step 14.2

Substitute any number, such as $- 6$ , from the interval $(- \infty, - 3)$ in the first derivative $\frac{x + 1}{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.2.1

Replace the variable $x$ with $- 6$ in the expression.

$f' (- 6) = \frac{(- 6) + 1}{{(- 6)}^{\frac{2}{3}} {((- 6) + 3)}^{\frac{1}{3}}}$

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Add $- 6$ and $1$ .

$f' (- 6) = \frac{- 5}{{(- 6)}^{\frac{2}{3}} {(- 6 + 3)}^{\frac{1}{3}}}$

Step 14.2.2.2

Add $- 6$ and $3$ .

$f' (- 6) = \frac{- 5}{{(- 6)}^{\frac{2}{3}} {(- 3)}^{\frac{1}{3}}}$

Step 14.2.2.3

Move the negative in front of the fraction.

$f' (- 6) = - \frac{5}{{(- 6)}^{\frac{2}{3}} {(- 3)}^{\frac{1}{3}}}$

Step 14.2.2.4

The final answer is $- \frac{5}{{(- 6)}^{\frac{2}{3}} {(- 3)}^{\frac{1}{3}}}$ .

$- \frac{5}{{(- 6)}^{\frac{2}{3}} {(- 3)}^{\frac{1}{3}}}$

Step 14.3

Substitute any number, such as $- 2$ , from the interval $(- 3, - 1)$ in the first derivative $\frac{x + 1}{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.3.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Add $- 2$ and $1$ .

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

Step 14.3.2.2

Simplify the denominator.

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Step 14.3.2.2.1

Add $- 2$ and $3$ .

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

Step 14.3.2.2.2

One to any power is one.

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

Step 14.3.2.3

Simplify the expression.

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Step 14.3.2.3.1

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

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

Step 14.3.2.3.2

Move the negative in front of the fraction.

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

Step 14.3.2.4

The final answer is $- \frac{1}{{(- 2)}^{\frac{2}{3}}}$ .

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

Step 14.4

Substitute any number, such as $- 0.5$ , from the interval $(- 1, 0)$ in the first derivative $\frac{x + 1}{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.4.1

Replace the variable $x$ with $- 0.5$ in the expression.

$f' (- 0.5) = \frac{(- 0.5) + 1}{{(- 0.5)}^{\frac{2}{3}} {((- 0.5) + 3)}^{\frac{1}{3}}}$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Add $- 0.5$ and $1$ .

$f' (- 0.5) = \frac{0.5}{{(- 0.5)}^{\frac{2}{3}} {(- 0.5 + 3)}^{\frac{1}{3}}}$

Step 14.4.2.2

Simplify the denominator.

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Step 14.4.2.2.1

Add $- 0.5$ and $3$ .

$f' (- 0.5) = \frac{0.5}{{(- 0.5)}^{\frac{2}{3}} \cdot {2.5}^{\frac{1}{3}}}$

Step 14.4.2.2.2

Raise $2.5$ to the power of $\frac{1}{3}$ .

$f' (- 0.5) = \frac{0.5}{{(- 0.5)}^{\frac{2}{3}} \cdot 1.3572088}$

Step 14.4.2.3

Move $1.3572088$ to the left of ${(- 0.5)}^{\frac{2}{3}}$ .

$f' (- 0.5) = \frac{0.5}{1.3572088 {(- 0.5)}^{\frac{2}{3}}}$

Step 14.4.2.4

The final answer is $\frac{0.5}{1.3572088 {(- 0.5)}^{\frac{2}{3}}}$ .

$\frac{0.5}{1.3572088 {(- 0.5)}^{\frac{2}{3}}}$

Step 14.5

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $\frac{x + 1}{x^{\frac{2}{3}} {(x + 3)}^{\frac{1}{3}}}$ to check if the result is negative or positive.

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Step 14.5.1

Replace the variable $x$ with $2$ in the expression.

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

Step 14.5.2

Simplify the result.

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Step 14.5.2.1

Add $2$ and $1$ .

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

Step 14.5.2.2

Add $2$ and $3$ .

$f' (2) = \frac{3}{2^{\frac{2}{3}} \cdot 5^{\frac{1}{3}}}$

Step 14.5.2.3

The final answer is $\frac{3}{2^{\frac{2}{3}} \cdot 5^{\frac{1}{3}}}$ .

$\frac{3}{2^{\frac{2}{3}} \cdot 5^{\frac{1}{3}}}$

Step 14.6

Since the first derivative changed signs from positive to negative around $x = - 3$ , then $x = - 3$ is a local maximum.

$x = - 3$ is a local maximum

Step 14.7

Since the first derivative changed signs from negative to positive around $x = - 1$ , then $x = - 1$ is a local minimum.

$x = - 1$ is a local minimum

Step 14.8

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.9

These are the local extrema for $f (x) = x^{\frac{1}{3}} {(x + 3)}^{\frac{2}{3}}$ .

$x = - 3$ is a local maximum

$x = - 1$ is a local minimum

$x = - 3$ is a local maximum

$x = - 1$ is a local minimum

Step 15