Calculus Examples

$f (x) = 3 \sqrt[3]{- x - 2} + 6$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $3 \sqrt[3]{- x - 2} + 6$ with respect to $x$ is $\frac{d}{d x} [3 \sqrt[3]{- x - 2}] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [3 \sqrt[3]{- x - 2}] + \frac{d}{d x} [6]$

Step 1.2

Evaluate $\frac{d}{d x} [3 \sqrt[3]{- x - 2}]$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{- x - 2}$ as ${(- x - 2)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [3 {(- x - 2)}^{\frac{1}{3}}] + \frac{d}{d x} [6]$

Step 1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 {(- x - 2)}^{\frac{1}{3}}$ with respect to $x$ is $3 \frac{d}{d x} [{(- x - 2)}^{\frac{1}{3}}]$ .

$3 \frac{d}{d x} [{(- x - 2)}^{\frac{1}{3}}] + \frac{d}{d x} [6]$

Step 1.2.3

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 $g (x) = - x - 2$ .

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Step 1.2.3.1

To apply the Chain Rule, set $u$ as $- x - 2$ .

$3 (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [- x - 2]) + \frac{d}{d x} [6]$

Step 1.2.3.2

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

$3 (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [- x - 2]) + \frac{d}{d x} [6]$

Step 1.2.3.3

Replace all occurrences of $u$ with $- x - 2$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} \frac{d}{d x} [- x - 2]) + \frac{d}{d x} [6]$

Step 1.2.4

By the Sum Rule, the derivative of $- x - 2$ with respect to $x$ is $\frac{d}{d x} [- x] + \frac{d}{d x} [- 2]$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} (\frac{d}{d x} [- x] + \frac{d}{d x} [- 2])) + \frac{d}{d x} [6]$

Step 1.2.5

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

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} (- \frac{d}{d x} [x] + \frac{d}{d x} [- 2])) + \frac{d}{d x} [6]$

Step 1.2.6

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

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} (- 1 \cdot 1 + \frac{d}{d x} [- 2])) + \frac{d}{d x} [6]$

Step 1.2.7

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 1.2.8

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

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 1.2.9

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

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 1.2.10

Combine the numerators over the common denominator.

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1 - 1 \cdot 3}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 1.2.11

Simplify the numerator.

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Step 1.2.11.1

Multiply $- 1$ by $3$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1 - 3}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 1.2.11.2

Subtract $3$ from $1$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{- 2}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 1.2.12

Move the negative in front of the fraction.

$3 (\frac{1}{3} {(- x - 2)}^{- \frac{2}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 1.2.13

Multiply $- 1$ by $1$ .

$3 (\frac{1}{3} {(- x - 2)}^{- \frac{2}{3}} (- 1 + 0)) + \frac{d}{d x} [6]$

Step 1.2.14

Add $- 1$ and $0$ .

$3 (\frac{1}{3} {(- x - 2)}^{- \frac{2}{3}} \cdot - 1) + \frac{d}{d x} [6]$

Step 1.2.15

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

$3 (\frac{{(- x - 2)}^{- \frac{2}{3}}}{3} \cdot - 1) + \frac{d}{d x} [6]$

Step 1.2.16

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

$3 \frac{{(- x - 2)}^{- \frac{2}{3}} \cdot - 1}{3} + \frac{d}{d x} [6]$

Step 1.2.17

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

$3 \frac{- 1 \cdot {(- x - 2)}^{- \frac{2}{3}}}{3} + \frac{d}{d x} [6]$

Step 1.2.18

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

$3 \frac{- {(- x - 2)}^{- \frac{2}{3}}}{3} + \frac{d}{d x} [6]$

Step 1.2.19

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

$3 \frac{- 1}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 1.2.20

Move the negative in front of the fraction.

$3 (- \frac{1}{3 {(- x - 2)}^{\frac{2}{3}}}) + \frac{d}{d x} [6]$

Step 1.2.21

Multiply $- 1$ by $3$ .

$- 3 \frac{1}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 1.2.22

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

$\frac{- 3}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 1.2.23

Factor $3$ out of $- 3$ .

$\frac{3 \cdot - 1}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 1.2.24

Cancel the common factors.

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Step 1.2.24.1

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

$\frac{3 \cdot - 1}{3 ({(- x - 2)}^{\frac{2}{3}})} + \frac{d}{d x} [6]$

Step 1.2.24.2

Cancel the common factor.

$\frac{3 \cdot - 1}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 1.2.24.3

Rewrite the expression.

$\frac{- 1}{{(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 1.2.25

Move the negative in front of the fraction.

$- \frac{1}{{(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

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

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

Step 1.3.2

Add $- \frac{1}{{(- x - 2)}^{\frac{2}{3}}}$ and $0$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2.3

Move the negative in front of the fraction.

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

Step 2.3

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 $g (x) = - x - 2$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $- x - 2$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u$ with $- x - 2$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

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

Step 2.7.2

Subtract $3$ from $- 2$ .

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

Step 2.8

Combine fractions.

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Step 2.8.1

Move the negative in front of the fraction.

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

Step 2.8.2

Combine ${(- x - 2)}^{- \frac{5}{3}}$ and $\frac{2}{3}$ .

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

Step 2.8.3

Simplify the expression.

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Step 2.8.3.1

Move $2$ to the left of ${(- x - 2)}^{- \frac{5}{3}}$ .

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

Step 2.8.3.2

Move ${(- x - 2)}^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8.3.3

Multiply $- 1$ by $- 1$ .

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

Step 2.8.3.4

Multiply $\frac{2}{3 {(- x - 2)}^{\frac{5}{3}}}$ by $1$ .

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

Step 2.9

By the Sum Rule, the derivative of $- x - 2$ with respect to $x$ is $\frac{d}{d x} [- x] + \frac{d}{d x} [- 2]$ .

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Multiply $- 1$ by $1$ .

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

Step 2.13

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f'' (x) = \frac{2}{3 {(- x - 2)}^{\frac{5}{3}}} \cdot (- 1 + 0) + \frac{1}{{(- x - 2)}^{\frac{2}{3}}} \cdot \frac{d}{d x} (- 1)$

Step 2.14

Combine fractions.

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Step 2.14.1

Add $- 1$ and $0$ .

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

Step 2.14.2

Combine $\frac{2}{3 {(- x - 2)}^{\frac{5}{3}}}$ and $- 1$ .

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

Step 2.14.3

Simplify the expression.

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Step 2.14.3.1

Multiply $2$ by $- 1$ .

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

Step 2.14.3.2

Move the negative in front of the fraction.

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

Step 2.15

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

$f'' (x) = - \frac{2}{3 {(- x - 2)}^{\frac{5}{3}}} + \frac{1}{{(- x - 2)}^{\frac{2}{3}}} \cdot 0$

Step 2.16

Simplify the expression.

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Step 2.16.1

Multiply $\frac{1}{{(- x - 2)}^{\frac{2}{3}}}$ by $0$ .

$f'' (x) = - \frac{2}{3 {(- x - 2)}^{\frac{5}{3}}} + 0$

Step 2.16.2

Add $- \frac{2}{3 {(- x - 2)}^{\frac{5}{3}}}$ and $0$ .

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

Step 3

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

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

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

By the Sum Rule, the derivative of $3 \sqrt[3]{- x - 2} + 6$ with respect to $x$ is $\frac{d}{d x} [3 \sqrt[3]{- x - 2}] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [3 \sqrt[3]{- x - 2}] + \frac{d}{d x} [6]$

Step 4.1.2

Evaluate $\frac{d}{d x} [3 \sqrt[3]{- x - 2}]$ .

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Step 4.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{- x - 2}$ as ${(- x - 2)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [3 {(- x - 2)}^{\frac{1}{3}}] + \frac{d}{d x} [6]$

Step 4.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 {(- x - 2)}^{\frac{1}{3}}$ with respect to $x$ is $3 \frac{d}{d x} [{(- x - 2)}^{\frac{1}{3}}]$ .

$3 \frac{d}{d x} [{(- x - 2)}^{\frac{1}{3}}] + \frac{d}{d x} [6]$

Step 4.1.2.3

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 $g (x) = - x - 2$ .

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Step 4.1.2.3.1

To apply the Chain Rule, set $u$ as $- x - 2$ .

$3 (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [- x - 2]) + \frac{d}{d x} [6]$

Step 4.1.2.3.2

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

$3 (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [- x - 2]) + \frac{d}{d x} [6]$

Step 4.1.2.3.3

Replace all occurrences of $u$ with $- x - 2$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} \frac{d}{d x} [- x - 2]) + \frac{d}{d x} [6]$

Step 4.1.2.4

By the Sum Rule, the derivative of $- x - 2$ with respect to $x$ is $\frac{d}{d x} [- x] + \frac{d}{d x} [- 2]$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} (\frac{d}{d x} [- x] + \frac{d}{d x} [- 2])) + \frac{d}{d x} [6]$

Step 4.1.2.5

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

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} (- \frac{d}{d x} [x] + \frac{d}{d x} [- 2])) + \frac{d}{d x} [6]$

Step 4.1.2.6

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

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} (- 1 \cdot 1 + \frac{d}{d x} [- 2])) + \frac{d}{d x} [6]$

Step 4.1.2.7

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 4.1.2.8

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

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 4.1.2.9

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

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 4.1.2.10

Combine the numerators over the common denominator.

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1 - 1 \cdot 3}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 4.1.2.11

Simplify the numerator.

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Step 4.1.2.11.1

Multiply $- 1$ by $3$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{1 - 3}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 4.1.2.11.2

Subtract $3$ from $1$ .

$3 (\frac{1}{3} {(- x - 2)}^{\frac{- 2}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 4.1.2.12

Move the negative in front of the fraction.

$3 (\frac{1}{3} {(- x - 2)}^{- \frac{2}{3}} (- 1 \cdot 1 + 0)) + \frac{d}{d x} [6]$

Step 4.1.2.13

Multiply $- 1$ by $1$ .

$3 (\frac{1}{3} {(- x - 2)}^{- \frac{2}{3}} (- 1 + 0)) + \frac{d}{d x} [6]$

Step 4.1.2.14

Add $- 1$ and $0$ .

$3 (\frac{1}{3} {(- x - 2)}^{- \frac{2}{3}} \cdot - 1) + \frac{d}{d x} [6]$

Step 4.1.2.15

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

$3 (\frac{{(- x - 2)}^{- \frac{2}{3}}}{3} \cdot - 1) + \frac{d}{d x} [6]$

Step 4.1.2.16

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

$3 \frac{{(- x - 2)}^{- \frac{2}{3}} \cdot - 1}{3} + \frac{d}{d x} [6]$

Step 4.1.2.17

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

$3 \frac{- 1 \cdot {(- x - 2)}^{- \frac{2}{3}}}{3} + \frac{d}{d x} [6]$

Step 4.1.2.18

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

$3 \frac{- {(- x - 2)}^{- \frac{2}{3}}}{3} + \frac{d}{d x} [6]$

Step 4.1.2.19

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

$3 \frac{- 1}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 4.1.2.20

Move the negative in front of the fraction.

$3 (- \frac{1}{3 {(- x - 2)}^{\frac{2}{3}}}) + \frac{d}{d x} [6]$

Step 4.1.2.21

Multiply $- 1$ by $3$ .

$- 3 \frac{1}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 4.1.2.22

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

$\frac{- 3}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 4.1.2.23

Factor $3$ out of $- 3$ .

$\frac{3 \cdot - 1}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 4.1.2.24

Cancel the common factors.

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Step 4.1.2.24.1

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

$\frac{3 \cdot - 1}{3 ({(- x - 2)}^{\frac{2}{3}})} + \frac{d}{d x} [6]$

Step 4.1.2.24.2

Cancel the common factor.

$\frac{3 \cdot - 1}{3 {(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 4.1.2.24.3

Rewrite the expression.

$\frac{- 1}{{(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 4.1.2.25

Move the negative in front of the fraction.

$- \frac{1}{{(- x - 2)}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 4.1.3

Differentiate using the Constant Rule.

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Step 4.1.3.1

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

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

Step 4.1.3.2

Add $- \frac{1}{{(- x - 2)}^{\frac{2}{3}}}$ and $0$ .

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$1 = 0$

Step 5.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

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

Step 6.2

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

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

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.

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

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

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

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

Step 6.3.2.2.1.2

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.2.2

Rewrite the expression.

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

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 $0$ .

${(- x - 2)}^{2} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Factor the left side of the equation.

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Step 6.3.3.1.1

Factor $- 1$ out of $- x - 2$ .

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Step 6.3.3.1.1.1

Factor $- 1$ out of $- x$ .

${(- (x) - 2)}^{2} = 0$

Step 6.3.3.1.1.2

Rewrite $- 2$ as $- 1 (2)$ .

${(- (x) - 1 \cdot 2)}^{2} = 0$

Step 6.3.3.1.1.3

Factor $- 1$ out of $- (x) - 1 (2)$ .

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

Step 6.3.3.1.2

Apply the product rule to $- (x + 2)$ .

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

Step 6.3.3.2

Divide each term in ${(- 1)}^{2} {(x + 2)}^{2} = 0$ by ${(- 1)}^{2}$ and simplify.

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Step 6.3.3.2.1

Divide each term in ${(- 1)}^{2} {(x + 2)}^{2} = 0$ by ${(- 1)}^{2}$ .

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

Step 6.3.3.2.2

Simplify the left side.

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Step 6.3.3.2.2.1

Cancel the common factor of ${(- 1)}^{2}$ .

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Step 6.3.3.2.2.1.1

Cancel the common factor.

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

Step 6.3.3.2.2.1.2

Divide ${(x + 2)}^{2}$ by $1$ .

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

Step 6.3.3.2.3

Simplify the right side.

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Step 6.3.3.2.3.1

Raise $- 1$ to the power of $2$ .

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

Step 6.3.3.2.3.2

Divide $0$ by $1$ .

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

Step 6.3.3.3

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 6.3.3.4

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7

Critical points to evaluate.

$x = - 2$

Step 8

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the expression.

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Step 9.1.1

Multiply $- 1$ by $- 2$ .

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

Step 9.1.2

Subtract $2$ from $2$ .

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

Step 9.1.3

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

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

Step 9.1.4

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

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

Step 9.2

Cancel the common factor of $3$ .

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Rewrite the expression.

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

Step 9.3

Simplify the expression.

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Step 9.3.1

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

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

Step 9.3.2

Multiply $3$ by $0$ .

$- \frac{2}{0}$

Step 9.3.3

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

Undefined

$- \frac{2}{0}$

Step 9.4

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

Undefined

Step 10

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

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Step 10.1

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

$(- \infty, - 2) \cup (- 2, \infty)$

Step 10.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - 2)$ in the first derivative $- \frac{1}{{(- x - 2)}^{\frac{2}{3}}}$ to check if the result is negative or positive.

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Step 10.2.1

Replace the variable $x$ with $- 4$ in the expression.

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

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Simplify the denominator.

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Step 10.2.2.1.1

Multiply $- 1$ by $- 4$ .

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

Step 10.2.2.1.2

Subtract $2$ from $4$ .

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

Step 10.2.2.2

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

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

Step 10.3

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

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Step 10.3.1

Replace the variable $x$ with $0$ in the expression.

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

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

Simplify the denominator.

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Step 10.3.2.1.1

Multiply $- 1$ by $0$ .

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

Step 10.3.2.1.2

Subtract $2$ from $0$ .

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

Step 10.3.2.2

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

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

Step 10.4

Since the first derivative did not change signs around $x = - 2$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 10.5

No local maxima or minima found for $f (x) = 3 \sqrt[3]{- x - 2} + 6$ .

No local maxima or minima

Step 11