Calculus Examples

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

Step 1

Write $\frac{\sqrt[3]{x}}{e^{x}} + 1$ as a function.

$f (x) = \frac{\sqrt[3]{x}}{e^{x}} + 1$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $\frac{\sqrt[3]{x}}{e^{x}} + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{\sqrt[3]{x}}{e^{x}}] + \frac{d}{d x} [1]$ .

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

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{\sqrt[3]{x}}{e^{x}}]$ .

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Step 2.1.2.1

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

$\frac{d}{d x} [\frac{x^{\frac{1}{3}}}{e^{x}}] + \frac{d}{d x} [1]$

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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

Step 2.1.2.7

Combine the numerators over the common denominator.

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

Step 2.1.2.8

Simplify the numerator.

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Step 2.1.2.8.1

Multiply $- 1$ by $3$ .

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

Step 2.1.2.8.2

Subtract $3$ from $1$ .

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

Step 2.1.2.9

Move the negative in front of the fraction.

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

Step 2.1.2.10

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

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

Step 2.1.2.11

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

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

Step 2.1.2.12

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

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

Step 2.1.2.13

Multiply the exponents in ${(e^{x})}^{2}$ .

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Step 2.1.2.13.1

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

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

Step 2.1.2.13.2

Move $2$ to the left of $x$ .

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

Step 2.1.2.14

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

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

Step 2.1.2.15

Combine.

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

Step 2.1.2.16

Apply the distributive property.

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

Step 2.1.2.17

Cancel the common factor of $3 x^{\frac{2}{3}}$ .

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Step 2.1.2.17.1

Cancel the common factor.

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

Step 2.1.2.17.2

Rewrite the expression.

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

Step 2.1.2.18

Multiply $- 1$ by $3$ .

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

Step 2.1.2.19

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

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

Step 2.1.2.20

Combine the numerators over the common denominator.

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

Step 2.1.2.21

Add $1$ and $2$ .

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

Step 2.1.2.22

Cancel the common factor of $3$ .

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Step 2.1.2.22.1

Cancel the common factor.

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

Step 2.1.2.22.2

Rewrite the expression.

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

Step 2.1.2.23

Simplify.

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

Step 2.1.3

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

$\frac{e^{x} - 3 x e^{x}}{3 x^{\frac{2}{3}} e^{2 x}} + 0$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Add $\frac{e^{x} - 3 x e^{x}}{3 x^{\frac{2}{3}} e^{2 x}}$ and $0$ .

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

Step 2.1.4.2

Reorder terms.

$\frac{- 3 e^{x} x + e^{x}}{3 e^{2 x} x^{\frac{2}{3}}}$

Step 2.1.4.3

Factor $e^{x}$ out of $- 3 e^{x} x + e^{x}$ .

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Step 2.1.4.3.1

Factor $e^{x}$ out of $- 3 e^{x} x$ .

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

Step 2.1.4.3.2

Multiply by $1$ .

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

Step 2.1.4.3.3

Factor $e^{x}$ out of $e^{x} (- 3 x) + e^{x} \cdot 1$ .

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

Step 2.1.4.4

Factor $e^{2 x}$ out of $e^{x} (- 3 x + 1)$ .

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

Step 2.1.4.5

Cancel the common factors.

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Step 2.1.4.5.1

Factor $e^{2 x}$ out of $3 e^{2 x} x^{\frac{2}{3}}$ .

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

Step 2.1.4.5.2

Cancel the common factor.

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

Step 2.1.4.5.3

Rewrite the expression.

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

Step 2.1.4.6

Factor $- 1$ out of $- 3 x$ .

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

Step 2.1.4.7

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

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

Step 2.1.4.8

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

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

Step 2.1.4.9

Rewrite $- (3 x - 1)$ as $- 1 (3 x - 1)$ .

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

Step 2.1.4.10

Move the negative in front of the fraction.

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

Step 2.2

Find the second derivative.

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Step 2.2.1

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

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

Step 2.2.2

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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Step 2.2.3.2.1

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

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

Step 2.2.3.2.2

Multiply $2$ by $2$ .

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} \frac{d}{d x} [e^{- x} (3 x - 1)] - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.4

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 $f (x) = e^{- x}$ and $g (x) = 3 x - 1$ .

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (e^{- x} \frac{d}{d x} [3 x - 1] + (3 x - 1) \frac{d}{d x} [e^{- x}]) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.5

Differentiate.

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Step 2.2.5.1

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (e^{- x} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [e^{- x}]) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.5.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (e^{- x} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [e^{- x}]) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.5.3

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

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (e^{- x} (3 \cdot 1 + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [e^{- x}]) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.5.4

Multiply $3$ by $1$ .

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (e^{- x} (3 + \frac{d}{d x} [- 1]) + (3 x - 1) \frac{d}{d x} [e^{- x}]) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.5.5

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

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (e^{- x} (3 + 0) + (3 x - 1) \frac{d}{d x} [e^{- x}]) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.5.6

Simplify the expression.

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Step 2.2.5.6.1

Add $3$ and $0$ .

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (e^{- x} \cdot 3 + (3 x - 1) \frac{d}{d x} [e^{- x}]) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.5.6.2

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

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (3 \cdot e^{- x} + (3 x - 1) \frac{d}{d x} [e^{- x}]) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.6

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

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Step 2.2.6.1

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

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (3 e^{- x} + (3 x - 1) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- x])) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.6.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (3 e^{- x} + (3 x - 1) (e^{u} \frac{d}{d x} [- x])) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.6.3

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

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (3 e^{- x} + (3 x - 1) (e^{- x} \frac{d}{d x} [- x])) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.7

Differentiate.

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Step 2.2.7.1

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

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (3 e^{- x} + (3 x - 1) e^{- x} (- \frac{d}{d x} [x])) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.7.2

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

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (3 e^{- x} + (3 x - 1) e^{- x} (- 1 \cdot 1)) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.7.3

Simplify the expression.

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Step 2.2.7.3.1

Multiply $- 1$ by $1$ .

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (3 e^{- x} + (3 x - 1) e^{- x} \cdot - 1) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.7.3.2

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

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (3 e^{- x} - 1 \cdot ((3 x - 1) e^{- x})) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.7.3.3

Rewrite $- 1 (3 x - 1)$ as $- (3 x - 1)$ .

$- \frac{1}{3} \cdot \frac{x^{\frac{2}{3}} (3 e^{- x} - (3 x - 1) e^{- x}) - e^{- x} (3 x - 1) \frac{d}{d x} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 2.2.7.4

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

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

Step 2.2.8

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

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

Step 2.2.9

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

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

Step 2.2.10

Combine the numerators over the common denominator.

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

Step 2.2.11

Simplify the numerator.

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Step 2.2.11.1

Multiply $- 1$ by $3$ .

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

Step 2.2.11.2

Subtract $3$ from $2$ .

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

Step 2.2.12

Move the negative in front of the fraction.

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

Step 2.2.13

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

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

Step 2.2.14

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

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

Step 2.2.15

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

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

Step 2.2.16

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

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

Step 2.2.17

Move $3$ to the left of $x^{\frac{4}{3}}$ .

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

Step 2.2.18

Simplify.

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Step 2.2.18.1

Apply the distributive property.

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

Step 2.2.18.2

Apply the distributive property.

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

Step 2.2.18.3

Apply the distributive property.

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

Step 2.2.18.4

Apply the distributive property.

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

Step 2.2.18.5

Simplify the numerator.

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Step 2.2.18.5.1

Simplify each term.

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Step 2.2.18.5.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.2.18.5.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.2.18.5.1.3

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

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Step 2.2.18.5.1.3.1

Move $x$ .

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

Step 2.2.18.5.1.3.2

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

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Step 2.2.18.5.1.3.2.1

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

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

Step 2.2.18.5.1.3.2.2

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

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

Step 2.2.18.5.1.3.3

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

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

Step 2.2.18.5.1.3.4

Combine the numerators over the common denominator.

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

Step 2.2.18.5.1.3.5

Add $3$ and $2$ .

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

Step 2.2.18.5.1.4

Multiply $- 1$ by $3$ .

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

Step 2.2.18.5.1.5

Rewrite using the commutative property of multiplication.

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} - 1 \cdot - 1 x^{\frac{2}{3}} e^{- x} - \frac{2 e^{- x}}{3 x^{\frac{1}{3}}} (3 x) - \frac{2 e^{- x}}{3 x^{\frac{1}{3}}} \cdot - 1}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.1.6

Multiply $- 1$ by $- 1$ .

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

Step 2.2.18.5.1.7

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

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

Step 2.2.18.5.1.8

Cancel the common factor of $3$ .

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Step 2.2.18.5.1.8.1

Move the leading negative in $- \frac{2 e^{- x}}{3 x^{\frac{1}{3}}}$ into the numerator.

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

Step 2.2.18.5.1.8.2

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

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

Step 2.2.18.5.1.8.3

Factor $3$ out of $3 x$ .

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

Step 2.2.18.5.1.8.4

Cancel the common factor.

Step 2.2.18.5.1.8.5

Rewrite the expression.

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + x^{\frac{2}{3}} e^{- x} + \frac{- 2 e^{- x}}{x^{\frac{1}{3}}} x - \frac{2 e^{- x}}{3 x^{\frac{1}{3}}} \cdot - 1}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.1.9

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

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + x^{\frac{2}{3}} e^{- x} + \frac{- 2 e^{- x} x}{x^{\frac{1}{3}}} - \frac{2 e^{- x}}{3 x^{\frac{1}{3}}} \cdot - 1}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.1.10

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

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + x^{\frac{2}{3}} e^{- x} - 2 e^{- x} x \cdot x^{- \frac{1}{3}} - \frac{2 e^{- x}}{3 x^{\frac{1}{3}}} \cdot - 1}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.1.11

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

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Step 2.2.18.5.1.11.1

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

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

Step 2.2.18.5.1.11.2

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

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Step 2.2.18.5.1.11.2.1

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

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

Step 2.2.18.5.1.11.2.2

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

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + x^{\frac{2}{3}} e^{- x} - 2 e^{- x} x^{- \frac{1}{3} + 1} - \frac{2 e^{- x}}{3 x^{\frac{1}{3}}} \cdot - 1}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.1.11.3

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

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + x^{\frac{2}{3}} e^{- x} - 2 e^{- x} x^{- \frac{1}{3} + \frac{3}{3}} - \frac{2 e^{- x}}{3 x^{\frac{1}{3}}} \cdot - 1}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.1.11.4

Combine the numerators over the common denominator.

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + x^{\frac{2}{3}} e^{- x} - 2 e^{- x} x^{\frac{- 1 + 3}{3}} - \frac{2 e^{- x}}{3 x^{\frac{1}{3}}} \cdot - 1}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.1.11.5

Add $- 1$ and $3$ .

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + x^{\frac{2}{3}} e^{- x} - 2 e^{- x} x^{\frac{2}{3}} - \frac{2 e^{- x}}{3 x^{\frac{1}{3}}} \cdot - 1}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.1.12

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

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Step 2.2.18.5.1.12.1

Multiply $- 1$ by $- 1$ .

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + x^{\frac{2}{3}} e^{- x} - 2 e^{- x} x^{\frac{2}{3}} + 1 \frac{2 e^{- x}}{3 x^{\frac{1}{3}}}}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.1.12.2

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

$- \frac{3 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + x^{\frac{2}{3}} e^{- x} - 2 e^{- x} x^{\frac{2}{3}} + \frac{2 e^{- x}}{3 x^{\frac{1}{3}}}}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.2

Add $3 x^{\frac{2}{3}} e^{- x}$ and $x^{\frac{2}{3}} e^{- x}$ .

$- \frac{4 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} - 2 e^{- x} x^{\frac{2}{3}} + \frac{2 e^{- x}}{3 x^{\frac{1}{3}}}}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.3

Subtract $2 e^{- x} x^{\frac{2}{3}}$ from $4 x^{\frac{2}{3}} e^{- x}$ .

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Step 2.2.18.5.3.1

Move $e^{- x}$ .

$- \frac{4 x^{\frac{2}{3}} e^{- x} - 2 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + \frac{2 e^{- x}}{3 x^{\frac{1}{3}}}}{3 x^{\frac{4}{3}}}$

Step 2.2.18.5.3.2

Subtract $2 x^{\frac{2}{3}} e^{- x}$ from $4 x^{\frac{2}{3}} e^{- x}$ .

$- \frac{2 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + \frac{2 e^{- x}}{3 x^{\frac{1}{3}}}}{3 x^{\frac{4}{3}}}$

Step 2.2.18.6

Combine terms.

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Step 2.2.18.6.1

Multiply $\frac{2 x^{\frac{2}{3}} e^{- x} - 3 x^{\frac{5}{3}} e^{- x} + \frac{2 e^{- x}}{3 x^{\frac{1}{3}}}}{3 x^{\frac{4}{3}}}$ by $\frac{3 x^{\frac{1}{3}}}{3 x^{\frac{1}{3}}}$ .

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

Step 2.2.18.6.2

Combine.

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

Step 2.2.18.6.3

Apply the distributive property.

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

Step 2.2.18.6.4

Cancel the common factor of $3 x^{\frac{1}{3}}$ .

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Step 2.2.18.6.4.1

Cancel the common factor.

Step 2.2.18.6.4.2

Rewrite the expression.

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

Step 2.2.18.6.5

Multiply $2$ by $3$ .

$- \frac{6 x^{\frac{1}{3}} (x^{\frac{2}{3}} e^{- x}) + 3 x^{\frac{1}{3}} (- 3 x^{\frac{5}{3}} e^{- x}) + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.6

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

$- \frac{6 x^{\frac{2}{3} + \frac{1}{3}} e^{- x} + 3 x^{\frac{1}{3}} (- 3 x^{\frac{5}{3}} e^{- x}) + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.7

Combine the numerators over the common denominator.

$- \frac{6 x^{\frac{2 + 1}{3}} e^{- x} + 3 x^{\frac{1}{3}} (- 3 x^{\frac{5}{3}} e^{- x}) + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.8

Add $2$ and $1$ .

$- \frac{6 x^{\frac{3}{3}} e^{- x} + 3 x^{\frac{1}{3}} (- 3 x^{\frac{5}{3}} e^{- x}) + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.9

Cancel the common factor of $3$ .

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Step 2.2.18.6.9.1

Cancel the common factor.

$- \frac{6 x^{\frac{3}{3}} e^{- x} + 3 x^{\frac{1}{3}} (- 3 x^{\frac{5}{3}} e^{- x}) + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.9.2

Rewrite the expression.

$- \frac{6 x^{1} e^{- x} + 3 x^{\frac{1}{3}} (- 3 x^{\frac{5}{3}} e^{- x}) + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.10

Simplify.

$- \frac{6 x e^{- x} + 3 x^{\frac{1}{3}} (- 3 x^{\frac{5}{3}} e^{- x}) + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.11

Multiply $- 3$ by $3$ .

$- \frac{6 x e^{- x} - 9 x^{\frac{1}{3}} (x^{\frac{5}{3}} e^{- x}) + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.12

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

$- \frac{6 x e^{- x} - 9 x^{\frac{5}{3} + \frac{1}{3}} e^{- x} + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.13

Combine the numerators over the common denominator.

$- \frac{6 x e^{- x} - 9 x^{\frac{5 + 1}{3}} e^{- x} + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.14

Add $5$ and $1$ .

$- \frac{6 x e^{- x} - 9 x^{\frac{6}{3}} e^{- x} + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.15

Cancel the common factor of $6$ and $3$ .

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Step 2.2.18.6.15.1

Factor $3$ out of $6$ .

$- \frac{6 x e^{- x} - 9 x^{\frac{3 \cdot 2}{3}} e^{- x} + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.15.2

Cancel the common factors.

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Step 2.2.18.6.15.2.1

Factor $3$ out of $3$ .

$- \frac{6 x e^{- x} - 9 x^{\frac{3 \cdot 2}{3 (1)}} e^{- x} + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.15.2.2

Cancel the common factor.

$- \frac{6 x e^{- x} - 9 x^{\frac{3 \cdot 2}{3 \cdot 1}} e^{- x} + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.15.2.3

Rewrite the expression.

$- \frac{6 x e^{- x} - 9 x^{\frac{2}{1}} e^{- x} + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.15.2.4

Divide $2$ by $1$ .

$- \frac{6 x e^{- x} - 9 x^{2} e^{- x} + 2 e^{- x}}{3 x^{\frac{1}{3}} (3 x^{\frac{4}{3}})}$

Step 2.2.18.6.16

Multiply $3$ by $3$ .

$- \frac{6 x e^{- x} - 9 x^{2} e^{- x} + 2 e^{- x}}{9 x^{\frac{1}{3}} x^{\frac{4}{3}}}$

Step 2.2.18.6.17

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

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Step 2.2.18.6.17.1

Move $x^{\frac{4}{3}}$ .

$- \frac{6 x e^{- x} - 9 x^{2} e^{- x} + 2 e^{- x}}{9 (x^{\frac{4}{3}} x^{\frac{1}{3}})}$

Step 2.2.18.6.17.2

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

$- \frac{6 x e^{- x} - 9 x^{2} e^{- x} + 2 e^{- x}}{9 x^{\frac{4}{3} + \frac{1}{3}}}$

Step 2.2.18.6.17.3

Combine the numerators over the common denominator.

$- \frac{6 x e^{- x} - 9 x^{2} e^{- x} + 2 e^{- x}}{9 x^{\frac{4 + 1}{3}}}$

Step 2.2.18.6.17.4

Add $4$ and $1$ .

$- \frac{6 x e^{- x} - 9 x^{2} e^{- x} + 2 e^{- x}}{9 x^{\frac{5}{3}}}$

Step 2.2.18.7

Simplify the numerator.

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Step 2.2.18.7.1

Factor $e^{- x}$ out of $6 x e^{- x} - 9 x^{2} e^{- x} + 2 e^{- x}$ .

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Step 2.2.18.7.1.1

Factor $e^{- x}$ out of $6 x e^{- x}$ .

$- \frac{e^{- x} (6 x) - 9 x^{2} e^{- x} + 2 e^{- x}}{9 x^{\frac{5}{3}}}$

Step 2.2.18.7.1.2

Factor $e^{- x}$ out of $- 9 x^{2} e^{- x}$ .

$- \frac{e^{- x} (6 x) + e^{- x} (- 9 x^{2}) + 2 e^{- x}}{9 x^{\frac{5}{3}}}$

Step 2.2.18.7.1.3

Factor $e^{- x}$ out of $2 e^{- x}$ .

$- \frac{e^{- x} (6 x) + e^{- x} (- 9 x^{2}) + e^{- x} \cdot 2}{9 x^{\frac{5}{3}}}$

Step 2.2.18.7.1.4

Factor $e^{- x}$ out of $e^{- x} (6 x) + e^{- x} (- 9 x^{2})$ .

$- \frac{e^{- x} (6 x - 9 x^{2}) + e^{- x} \cdot 2}{9 x^{\frac{5}{3}}}$

Step 2.2.18.7.1.5

Factor $e^{- x}$ out of $e^{- x} (6 x - 9 x^{2}) + e^{- x} \cdot 2$ .

$- \frac{e^{- x} (6 x - 9 x^{2} + 2)}{9 x^{\frac{5}{3}}}$

Step 2.2.18.7.2

Reorder terms.

$- \frac{e^{- x} (- 9 x^{2} + 6 x + 2)}{9 x^{\frac{5}{3}}}$

Step 2.2.18.8

Factor $- 1$ out of $- 9 x^{2}$ .

$- \frac{e^{- x} (- (9 x^{2}) + 6 x + 2)}{9 x^{\frac{5}{3}}}$

Step 2.2.18.9

Factor $- 1$ out of $6 x$ .

$- \frac{e^{- x} (- (9 x^{2}) - (- 6 x) + 2)}{9 x^{\frac{5}{3}}}$

Step 2.2.18.10

Factor $- 1$ out of $- (9 x^{2}) - (- 6 x)$ .

$- \frac{e^{- x} (- (9 x^{2} - 6 x) + 2)}{9 x^{\frac{5}{3}}}$

Step 2.2.18.11

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

$- \frac{e^{- x} (- (9 x^{2} - 6 x) - 1 (- 2))}{9 x^{\frac{5}{3}}}$

Step 2.2.18.12

Factor $- 1$ out of $- (9 x^{2} - 6 x) - 1 (- 2)$ .

$- \frac{e^{- x} (- (9 x^{2} - 6 x - 2))}{9 x^{\frac{5}{3}}}$

Step 2.2.18.13

Rewrite $- (9 x^{2} - 6 x - 2)$ as $- 1 (9 x^{2} - 6 x - 2)$ .

$- \frac{e^{- x} (- 1 (9 x^{2} - 6 x - 2))}{9 x^{\frac{5}{3}}}$

Step 2.2.18.14

Move the negative in front of the fraction.

$- - \frac{e^{- x} (9 x^{2} - 6 x - 2)}{9 x^{\frac{5}{3}}}$

Step 2.2.18.15

Multiply $- 1$ by $- 1$ .

$1 \frac{e^{- x} (9 x^{2} - 6 x - 2)}{9 x^{\frac{5}{3}}}$

Step 2.2.18.16

Multiply $\frac{e^{- x} (9 x^{2} - 6 x - 2)}{9 x^{\frac{5}{3}}}$ by $1$ .

$f'' (x) = \frac{e^{- x} (9 x^{2} - 6 x - 2)}{9 x^{\frac{5}{3}}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{e^{- x} (9 x^{2} - 6 x - 2)}{9 x^{\frac{5}{3}}}$ .

$\frac{e^{- x} (9 x^{2} - 6 x - 2)}{9 x^{\frac{5}{3}}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{e^{- x} (9 x^{2} - 6 x - 2)}{9 x^{\frac{5}{3}}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{e^{- x} (9 x^{2} - 6 x - 2)}{9 x^{\frac{5}{3}}} = 0$

Step 3.2

Set the numerator equal to zero.

$e^{- x} (9 x^{2} - 6 x - 2) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

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

$e^{- x} = 0$

$9 x^{2} - 6 x - 2 = 0$

Step 3.3.2

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

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Step 3.3.2.1

Set $e^{- x}$ equal to $0$ .

$e^{- x} = 0$

Step 3.3.2.2

Solve $e^{- x} = 0$ for $x$ .

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Step 3.3.2.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- x}) = \ln (0)$

Step 3.3.2.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 3.3.2.2.3

There is no solution for $e^{- x} = 0$

No solution

Step 3.3.3

Set $9 x^{2} - 6 x - 2$ equal to $0$ and solve for $x$ .

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Step 3.3.3.1

Set $9 x^{2} - 6 x - 2$ equal to $0$ .

$9 x^{2} - 6 x - 2 = 0$

Step 3.3.3.2

Solve $9 x^{2} - 6 x - 2 = 0$ for $x$ .

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Step 3.3.3.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3.3.2.2

Substitute the values $a = 9$ , $b = - 6$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (9 \cdot - 2)}}{2 \cdot 9}$

Step 3.3.3.2.3

Simplify.

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Step 3.3.3.2.3.1

Simplify the numerator.

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Step 3.3.3.2.3.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Step 3.3.3.2.3.1.2

Multiply $- 4 \cdot 9 \cdot - 2$ .

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Step 3.3.3.2.3.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{6 \pm \sqrt{36 - 36 \cdot - 2}}{2 \cdot 9}$

Step 3.3.3.2.3.1.2.2

Multiply $- 36$ by $- 2$ .

$x = \frac{6 \pm \sqrt{36 + 72}}{2 \cdot 9}$

Step 3.3.3.2.3.1.3

Add $36$ and $72$ .

$x = \frac{6 \pm \sqrt{108}}{2 \cdot 9}$

Step 3.3.3.2.3.1.4

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 3.3.3.2.3.1.4.1

Factor $36$ out of $108$ .

$x = \frac{6 \pm \sqrt{36 (3)}}{2 \cdot 9}$

Step 3.3.3.2.3.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{6 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 9}$

Step 3.3.3.2.3.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 6 \sqrt{3}}{2 \cdot 9}$

Step 3.3.3.2.3.2

Multiply $2$ by $9$ .

$x = \frac{6 \pm 6 \sqrt{3}}{18}$

Step 3.3.3.2.3.3

Simplify $\frac{6 \pm 6 \sqrt{3}}{18}$ .

$x = \frac{1 \pm \sqrt{3}}{3}$

Step 3.3.3.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.3.3.2.4.1

Simplify the numerator.

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Step 3.3.3.2.4.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Step 3.3.3.2.4.1.2

Multiply $- 4 \cdot 9 \cdot - 2$ .

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Step 3.3.3.2.4.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{6 \pm \sqrt{36 - 36 \cdot - 2}}{2 \cdot 9}$

Step 3.3.3.2.4.1.2.2

Multiply $- 36$ by $- 2$ .

$x = \frac{6 \pm \sqrt{36 + 72}}{2 \cdot 9}$

Step 3.3.3.2.4.1.3

Add $36$ and $72$ .

$x = \frac{6 \pm \sqrt{108}}{2 \cdot 9}$

Step 3.3.3.2.4.1.4

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 3.3.3.2.4.1.4.1

Factor $36$ out of $108$ .

$x = \frac{6 \pm \sqrt{36 (3)}}{2 \cdot 9}$

Step 3.3.3.2.4.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{6 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 9}$

Step 3.3.3.2.4.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 6 \sqrt{3}}{2 \cdot 9}$

Step 3.3.3.2.4.2

Multiply $2$ by $9$ .

$x = \frac{6 \pm 6 \sqrt{3}}{18}$

Step 3.3.3.2.4.3

Simplify $\frac{6 \pm 6 \sqrt{3}}{18}$ .

$x = \frac{1 \pm \sqrt{3}}{3}$

Step 3.3.3.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{3}}{3}$

Step 3.3.3.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.3.3.2.5.1

Simplify the numerator.

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Step 3.3.3.2.5.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Step 3.3.3.2.5.1.2

Multiply $- 4 \cdot 9 \cdot - 2$ .

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Step 3.3.3.2.5.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{6 \pm \sqrt{36 - 36 \cdot - 2}}{2 \cdot 9}$

Step 3.3.3.2.5.1.2.2

Multiply $- 36$ by $- 2$ .

$x = \frac{6 \pm \sqrt{36 + 72}}{2 \cdot 9}$

Step 3.3.3.2.5.1.3

Add $36$ and $72$ .

$x = \frac{6 \pm \sqrt{108}}{2 \cdot 9}$

Step 3.3.3.2.5.1.4

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 3.3.3.2.5.1.4.1

Factor $36$ out of $108$ .

$x = \frac{6 \pm \sqrt{36 (3)}}{2 \cdot 9}$

Step 3.3.3.2.5.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{6 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 9}$

Step 3.3.3.2.5.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 6 \sqrt{3}}{2 \cdot 9}$

Step 3.3.3.2.5.2

Multiply $2$ by $9$ .

$x = \frac{6 \pm 6 \sqrt{3}}{18}$

Step 3.3.3.2.5.3

Simplify $\frac{6 \pm 6 \sqrt{3}}{18}$ .

$x = \frac{1 \pm \sqrt{3}}{3}$

Step 3.3.3.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{3}}{3}$

Step 3.3.3.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{3}}{3}, \frac{1 - \sqrt{3}}{3}$

Step 3.3.4

The final solution is all the values that make $e^{- x} (9 x^{2} - 6 x - 2) = 0$ true.

$x = \frac{1 + \sqrt{3}}{3}, \frac{1 - \sqrt{3}}{3}$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $0.9106836$ in $f (x) = \frac{\sqrt[3]{x}}{e^{x}} + 1$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $0.9106836$ in the expression.

$f (0.9106836) = \frac{\sqrt[3]{0.9106836}}{e^{0.9106836}} + 1$

Step 4.1.2

The final answer is $\frac{\sqrt[3]{0.9106836}}{e^{0.9106836}} + 1$ .

$\frac{\sqrt[3]{0.9106836}}{e^{0.9106836}} + 1$

Step 4.2

The point found by substituting $0.9106836$ in $f (x) = \frac{\sqrt[3]{x}}{e^{x}} + 1$ is $(0.9106836, \frac{\sqrt[3]{0.9106836}}{e^{0.9106836}} + 1)$ . This point can be an inflection point.

$(0.9106836, \frac{\sqrt[3]{0.9106836}}{e^{0.9106836}} + 1)$

Step 4.3

Substitute $- 0.24401693$ in $f (x) = \frac{\sqrt[3]{x}}{e^{x}} + 1$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $- 0.24401693$ in the expression.

$f (- 0.24401693) = \frac{\sqrt[3]{- 0.24401693}}{e^{- 0.24401693}} + 1$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Move $e^{- 0.24401693}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$f (- 0.24401693) = \sqrt[3]{- 0.24401693} e^{0.24401693} + 1$

Step 4.3.2.1.2

Rewrite $- 0.24401693$ as ${(- 1)}^{3} \cdot 0.24401693$ .

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Step 4.3.2.1.2.1

Rewrite.

$f (- 0.24401693) = \sqrt[3]{- 1 \cdot 0.24401693} e^{0.24401693} + 1$

Step 4.3.2.1.2.2

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

$f (- 0.24401693) = \sqrt[3]{{(- 1)}^{3} \cdot 0.24401693} e^{0.24401693} + 1$

Step 4.3.2.1.3

Pull terms out from under the radical.

$f (- 0.24401693) = - 1 \sqrt[3]{0.24401693} e^{0.24401693} + 1$

Step 4.3.2.1.4

Rewrite $- 1 \sqrt[3]{0.24401693}$ as $- \sqrt[3]{0.24401693}$ .

$f (- 0.24401693) = - \sqrt[3]{0.24401693} e^{0.24401693} + 1$

Step 4.3.2.2

The final answer is $- \sqrt[3]{0.24401693} e^{0.24401693} + 1$ .

$- \sqrt[3]{0.24401693} e^{0.24401693} + 1$

Step 4.4

The point found by substituting $- 0.24401693$ in $f (x) = \frac{\sqrt[3]{x}}{e^{x}} + 1$ is $(- 0.24401693, - \sqrt[3]{0.24401693} e^{0.24401693} + 1)$ . This point can be an inflection point.

$(- 0.24401693, - \sqrt[3]{0.24401693} e^{0.24401693} + 1)$

Step 4.5

Determine the points that could be inflection points.

$(0.9106836, \frac{\sqrt[3]{0.9106836}}{e^{0.9106836}} + 1), (- 0.24401693, - \sqrt[3]{0.24401693} e^{0.24401693} + 1)$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - 0.24401693) \cup (- 0.24401693, 0.9106836) \cup (0.9106836, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 0.24401693)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.34401693$ in the expression.

$f'' (- 0.34401693) = \frac{e^{- (- 0.34401693)} (9 {(- 0.34401693)}^{2} - 6 \cdot - 0.34401693 - 2)}{9 {(- 0.34401693)}^{\frac{5}{3}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Raise $- 0.34401693$ to the power of $2$ .

$f'' (- 0.34401693) = \frac{e^{- (- 0.34401693)} (9 \cdot 0.11834765 - 6 \cdot - 0.34401693 - 2)}{9 {(- 0.34401693)}^{\frac{5}{3}}}$

Step 6.2.1.2

Multiply $9$ by $0.11834765$ .

$f'' (- 0.34401693) = \frac{e^{- (- 0.34401693)} (1.06512886 - 6 \cdot - 0.34401693 - 2)}{9 {(- 0.34401693)}^{\frac{5}{3}}}$

Step 6.2.1.3

Multiply $- 6$ by $- 0.34401693$ .

$f'' (- 0.34401693) = \frac{e^{- (- 0.34401693)} (1.06512886 + 2.06410161 - 2)}{9 {(- 0.34401693)}^{\frac{5}{3}}}$

Step 6.2.1.4

Add $1.06512886$ and $2.06410161$ .

$f'' (- 0.34401693) = \frac{e^{- (- 0.34401693)} (3.12923048 - 2)}{9 {(- 0.34401693)}^{\frac{5}{3}}}$

Step 6.2.1.5

Subtract $2$ from $3.12923048$ .

$f'' (- 0.34401693) = \frac{e^{- (- 0.34401693)} \cdot 1.12923048}{9 {(- 0.34401693)}^{\frac{5}{3}}}$

Step 6.2.1.6

Multiply $- 1$ by $- 0.34401693$ .

$f'' (- 0.34401693) = \frac{e^{0.34401693} \cdot 1.12923048}{9 {(- 0.34401693)}^{\frac{5}{3}}}$

Step 6.2.2

Multiply $e^{0.34401693}$ by $1.12923048$ .

$f'' (- 0.34401693) = \frac{1.59289537}{9 {(- 0.34401693)}^{\frac{5}{3}}}$

Step 6.2.3

The final answer is $\frac{1.59289537}{9 {(- 0.34401693)}^{\frac{5}{3}}}$ .

$\frac{1.59289537}{9 {(- 0.34401693)}^{\frac{5}{3}}}$

Step 6.3

At $- 0.34401693$ , the second derivative is $- 1.04788036$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 0.24401693)$

Decreasing on $(- \infty, - 0.24401693)$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(- 0.24401693, 0.9106836)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.\overline{3}$ in the expression.

$f'' (0.\overline{3}) = \frac{e^{- (0.\overline{3})} (9 {(0.\overline{3})}^{2} - 6 \cdot 0.\overline{3} - 2)}{9 {(0.\overline{3})}^{\frac{5}{3}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Move $e^{- (0.\overline{3})}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (0.\overline{3}) = \frac{9 {(0.\overline{3})}^{2} - 6 \cdot 0.\overline{3} - 2}{9 {(0.\overline{3})}^{\frac{5}{3}} e^{0.\overline{3}}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Raise $0.\overline{3}$ to the power of $2$ .

$f'' (0.\overline{3}) = \frac{9 \cdot 0.\overline{1} - 6 \cdot 0.\overline{3} - 2}{9 \cdot ({0.\overline{3}}^{\frac{5}{3}} e^{0.\overline{3}})}$

Step 7.2.2.2

Multiply $9$ by $0.\overline{1}$ .

$f'' (0.\overline{3}) = \frac{1 - 6 \cdot 0.\overline{3} - 2}{9 \cdot ({0.\overline{3}}^{\frac{5}{3}} e^{0.\overline{3}})}$

Step 7.2.2.3

Multiply $- 6$ by $0.\overline{3}$ .

$f'' (0.\overline{3}) = \frac{1 - 2 - 2}{9 \cdot ({0.\overline{3}}^{\frac{5}{3}} e^{0.\overline{3}})}$

Step 7.2.2.4

Subtract $2$ from $1$ .

$f'' (0.\overline{3}) = \frac{- 1 - 2}{9 \cdot ({0.\overline{3}}^{\frac{5}{3}} e^{0.\overline{3}})}$

Step 7.2.2.5

Subtract $2$ from $- 1$ .

$f'' (0.\overline{3}) = \frac{- 3}{9 \cdot ({0.\overline{3}}^{\frac{5}{3}} e^{0.\overline{3}})}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Raise $0.\overline{3}$ to the power of $\frac{5}{3}$ .

$f'' (0.\overline{3}) = \frac{- 3}{9 \cdot (0.16024995 e^{0.\overline{3}})}$

Step 7.2.3.2

Combine exponents.

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Step 7.2.3.2.1

Multiply $9$ by $0.16024995$ .

$f'' (0.\overline{3}) = \frac{- 3}{1.44224957 e^{0.\overline{3}}}$

Step 7.2.3.2.2

Multiply $1.44224957$ by $e^{0.\overline{3}}$ .

$f'' (0.\overline{3}) = \frac{- 3}{2.01282142}$

Step 7.2.4

Divide $- 3$ by $2.01282142$ .

$f'' (0.\overline{3}) = - 1.49044518$

Step 7.2.5

The final answer is $- 1.49044518$ .

$- 1.49044518$

Step 7.3

At $0.\overline{3}$ , the second derivative is $- 1.49044518$ . Since this is negative, the second derivative is decreasing on the interval $(- 0.24401693, 0.9106836)$

Decreasing on $(- 0.24401693, 0.9106836)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(0.9106836, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $1.0106836$ in the expression.

$f'' (1.0106836) = \frac{e^{- (1.0106836)} (9 {(1.0106836)}^{2} - 6 \cdot 1.0106836 - 2)}{9 {(1.0106836)}^{\frac{5}{3}}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Move $e^{- (1.0106836)}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (1.0106836) = \frac{9 {(1.0106836)}^{2} - 6 \cdot 1.0106836 - 2}{9 {(1.0106836)}^{\frac{5}{3}} e^{1.0106836}}$

Step 8.2.2

Simplify the numerator.

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Step 8.2.2.1

Raise $1.0106836$ to the power of $2$ .

$f'' (1.0106836) = \frac{9 \cdot 1.02148134 - 6 \cdot 1.0106836 - 2}{9 \cdot ({1.0106836}^{\frac{5}{3}} e^{1.0106836})}$

Step 8.2.2.2

Multiply $9$ by $1.02148134$ .

$f'' (1.0106836) = \frac{9.19333209 - 6 \cdot 1.0106836 - 2}{9 \cdot ({1.0106836}^{\frac{5}{3}} e^{1.0106836})}$

Step 8.2.2.3

Multiply $- 6$ by $1.0106836$ .

$f'' (1.0106836) = \frac{9.19333209 - 6.06410161 - 2}{9 \cdot ({1.0106836}^{\frac{5}{3}} e^{1.0106836})}$

Step 8.2.2.4

Subtract $6.06410161$ from $9.19333209$ .

$f'' (1.0106836) = \frac{3.12923048 - 2}{9 \cdot ({1.0106836}^{\frac{5}{3}} e^{1.0106836})}$

Step 8.2.2.5

Subtract $2$ from $3.12923048$ .

$f'' (1.0106836) = \frac{1.12923048}{9 \cdot ({1.0106836}^{\frac{5}{3}} e^{1.0106836})}$

Step 8.2.3

Simplify the denominator.

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Step 8.2.3.1

Raise $1.0106836$ to the power of $\frac{5}{3}$ .

$f'' (1.0106836) = \frac{1.12923048}{9 \cdot (1.01786933 e^{1.0106836})}$

Step 8.2.3.2

Combine exponents.

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Step 8.2.3.2.1

Multiply $9$ by $1.01786933$ .

$f'' (1.0106836) = \frac{1.12923048}{9.16082405 e^{1.0106836}}$

Step 8.2.3.2.2

Multiply $9.16082405$ by $e^{1.0106836}$ .

$f'' (1.0106836) = \frac{1.12923048}{25.16916766}$

Step 8.2.4

Divide $1.12923048$ by $25.16916766$ .

$f'' (1.0106836) = 0.04486562$

Step 8.2.5

The final answer is $0.04486562$ .

$0.04486562$

Step 8.3

At $1.0106836$ , the second derivative is $0.04486562$ . Since this is positive, the second derivative is increasing on the interval $(0.9106836, \infty)$ .

Increasing on $(0.9106836, \infty)$ since $f'' (x) > 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(0.9106836, \frac{\sqrt[3]{0.9106836}}{e^{0.9106836}} + 1)$ .

$(0.9106836, \frac{\sqrt[3]{0.9106836}}{e^{0.9106836}} + 1)$

Step 10