Calculus Examples

$9 \sqrt{x} e^{- x}$

Step 1

Write $9 \sqrt{x} e^{- x}$ as a function.

$f (x) = 9 \sqrt{x} e^{- x}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1.1.1

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

$\frac{d}{d x} [9 x^{\frac{1}{2}} e^{- x}]$

Step 2.1.1.1.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{1}{2}} e^{- x}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{1}{2}} e^{- x}]$ .

$9 \frac{d}{d x} [x^{\frac{1}{2}} e^{- x}]$

Step 2.1.1.2

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

$9 (x^{\frac{1}{2}} \frac{d}{d x} [e^{- x}] + e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

$9 (x^{\frac{1}{2}} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- x]) + e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.1.1.3.2

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

$9 (x^{\frac{1}{2}} (e^{u} \frac{d}{d x} [- x]) + e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.1.1.3.3

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

$9 (x^{\frac{1}{2}} (e^{- x} \frac{d}{d x} [- x]) + e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.1.1.4

Differentiate.

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Step 2.1.1.4.1

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

$9 (x^{\frac{1}{2}} (e^{- x} (- \frac{d}{d x} [x])) + e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.1.1.4.2

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

$9 (x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1)) + e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.1.1.4.3

Simplify the expression.

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Step 2.1.1.4.3.1

Multiply $- 1$ by $1$ .

$9 (x^{\frac{1}{2}} (e^{- x} \cdot - 1) + e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.1.1.4.3.2

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

$9 (x^{\frac{1}{2}} (- 1 \cdot e^{- x}) + e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.1.1.4.3.3

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

$9 (x^{\frac{1}{2}} (- e^{- x}) + e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.1.1.4.4

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

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

Step 2.1.1.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.1.1.6

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

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

Step 2.1.1.7

Combine the numerators over the common denominator.

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

Step 2.1.1.8

Simplify the numerator.

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Step 2.1.1.8.1

Multiply $- 1$ by $2$ .

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

Step 2.1.1.8.2

Subtract $2$ from $1$ .

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

Step 2.1.1.9

Move the negative in front of the fraction.

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

Step 2.1.1.10

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

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

Step 2.1.1.11

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

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

Step 2.1.1.12

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

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

Step 2.1.1.13

Simplify.

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Step 2.1.1.13.1

Apply the distributive property.

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

Step 2.1.1.13.2

Combine terms.

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Step 2.1.1.13.2.1

Multiply $- 1$ by $9$ .

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

Step 2.1.1.13.2.2

Combine $9$ and $\frac{e^{- x}}{2 x^{\frac{1}{2}}}$ .

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

Step 2.1.1.13.3

Reorder terms.

$f' (x) = - 9 e^{- x} x^{\frac{1}{2}} + \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $- 9 e^{- x} x^{\frac{1}{2}} + \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{d}{d x} [- 9 e^{- x} x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$ .

$\frac{d}{d x} [- 9 e^{- x} x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2

Evaluate $\frac{d}{d x} [- 9 e^{- x} x^{\frac{1}{2}}]$ .

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Step 2.1.2.2.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 e^{- x} x^{\frac{1}{2}}$ with respect to $x$ is $- 9 \frac{d}{d x} [e^{- x} x^{\frac{1}{2}}]$ .

$- 9 \frac{d}{d x} [e^{- x} x^{\frac{1}{2}}] + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.2

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 $g (x) = x^{\frac{1}{2}}$ .

$- 9 (e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}] + x^{\frac{1}{2}} \frac{d}{d x} [e^{- x}]) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.3

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

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1}{2} - 1}) + x^{\frac{1}{2}} \frac{d}{d x} [e^{- x}]) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.4

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

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Step 2.1.2.2.4.1

To apply the Chain Rule, set $u_{1}$ as $- x$ .

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1}{2} - 1}) + x^{\frac{1}{2}} (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- x])) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1}{2} - 1}) + x^{\frac{1}{2}} (e^{u_{1}} \frac{d}{d x} [- x])) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.4.3

Replace all occurrences of $u_{1}$ with $- x$ .

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1}{2} - 1}) + x^{\frac{1}{2}} (e^{- x} \frac{d}{d x} [- x])) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.5

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

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1}{2} - 1}) + x^{\frac{1}{2}} (e^{- x} (- \frac{d}{d x} [x]))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.6

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

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1}{2} - 1}) + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.8

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

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.9

Combine the numerators over the common denominator.

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.10

Simplify the numerator.

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Step 2.1.2.2.10.1

Multiply $- 1$ by $2$ .

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{1 - 2}{2}}) + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.10.2

Subtract $2$ from $1$ .

$- 9 (e^{- x} (\frac{1}{2} x^{\frac{- 1}{2}}) + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.11

Move the negative in front of the fraction.

$- 9 (e^{- x} (\frac{1}{2} x^{- \frac{1}{2}}) + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.12

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

$- 9 (e^{- x} \frac{x^{- \frac{1}{2}}}{2} + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.13

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

$- 9 (\frac{e^{- x} x^{- \frac{1}{2}}}{2} + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.14

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

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1))) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.15

Multiply $- 1$ by $1$ .

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (e^{- x} \cdot - 1)) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.16

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

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (- 1 \cdot e^{- x})) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.2.17

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (- e^{- x})) + \frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$

Step 2.1.2.3

Evaluate $\frac{d}{d x} [\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}]$ .

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Step 2.1.2.3.1

Since $\frac{9}{2}$ is constant with respect to $x$ , the derivative of $\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{9}{2} \frac{d}{d x} [\frac{e^{- x}}{x^{\frac{1}{2}}}]$ .

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (- e^{- x})) + \frac{9}{2} \frac{d}{d x} [\frac{e^{- x}}{x^{\frac{1}{2}}}]$

Step 2.1.2.3.2

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

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (- e^{- x})) + \frac{9}{2} \cdot \frac{x^{\frac{1}{2}} \frac{d}{d x} [e^{- x}] - e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}]}{{(x^{\frac{1}{2}})}^{2}}$

Step 2.1.2.3.3

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

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Step 2.1.2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $- x$ .

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (- e^{- x})) + \frac{9}{2} \cdot \frac{x^{\frac{1}{2}} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [- x]) - e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}]}{{(x^{\frac{1}{2}})}^{2}}$

Step 2.1.2.3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (- e^{- x})) + \frac{9}{2} \cdot \frac{x^{\frac{1}{2}} (e^{u_{2}} \frac{d}{d x} [- x]) - e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}]}{{(x^{\frac{1}{2}})}^{2}}$

Step 2.1.2.3.3.3

Replace all occurrences of $u_{2}$ with $- x$ .

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (- e^{- x})) + \frac{9}{2} \cdot \frac{x^{\frac{1}{2}} (e^{- x} \frac{d}{d x} [- x]) - e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}]}{{(x^{\frac{1}{2}})}^{2}}$

Step 2.1.2.3.4

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

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (- e^{- x})) + \frac{9}{2} \cdot \frac{x^{\frac{1}{2}} (e^{- x} (- \frac{d}{d x} [x])) - e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}]}{{(x^{\frac{1}{2}})}^{2}}$

Step 2.1.2.3.5

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

$- 9 (\frac{e^{- x}}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (- e^{- x})) + \frac{9}{2} \cdot \frac{x^{\frac{1}{2}} (e^{- x} (- 1 \cdot 1)) - e^{- x} \frac{d}{d x} [x^{\frac{1}{2}}]}{{(x^{\frac{1}{2}})}^{2}}$

Step 2.1.2.3.6

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

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

Step 2.1.2.3.7

Multiply $- 1$ by $1$ .

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

Step 2.1.2.3.8

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

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

Step 2.1.2.3.9

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

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

Step 2.1.2.3.10

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.1.2.3.11

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

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

Step 2.1.2.3.12

Combine the numerators over the common denominator.

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

Step 2.1.2.3.13

Simplify the numerator.

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Step 2.1.2.3.13.1

Multiply $- 1$ by $2$ .

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

Step 2.1.2.3.13.2

Subtract $2$ from $1$ .

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

Step 2.1.2.3.14

Move the negative in front of the fraction.

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

Step 2.1.2.3.15

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

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

Step 2.1.2.3.16

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

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

Step 2.1.2.3.17

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

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

Step 2.1.2.3.18

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

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Step 2.1.2.3.18.1

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

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

Step 2.1.2.3.18.2

Cancel the common factor of $2$ .

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Step 2.1.2.3.18.2.1

Cancel the common factor.

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

Step 2.1.2.3.18.2.2

Rewrite the expression.

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

Step 2.1.2.3.19

Simplify.

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

Step 2.1.2.3.20

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

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

Step 2.1.2.4

Simplify.

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Step 2.1.2.4.1

Apply the distributive property.

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

Step 2.1.2.4.2

Apply the distributive property.

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

Step 2.1.2.4.3

Combine terms.

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Step 2.1.2.4.3.1

Combine $- 9$ and $\frac{e^{- x}}{2 x^{\frac{1}{2}}}$ .

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

Step 2.1.2.4.3.2

Move the negative in front of the fraction.

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

Step 2.1.2.4.3.3

Multiply $- 1$ by $- 9$ .

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

Step 2.1.2.4.3.4

Multiply $- 1$ by $9$ .

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

Step 2.1.2.4.3.5

Multiply $- 1$ by $9$ .

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

Step 2.1.2.4.3.6

Combine $- 9$ and $\frac{e^{- x}}{2 x^{\frac{1}{2}}}$ .

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

Step 2.1.2.4.3.7

Move the negative in front of the fraction.

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

Step 2.1.2.4.3.8

To write $- \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}} \cdot \frac{x}{x} + \frac{- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}}{2 x}$

Step 2.1.2.4.3.9

To write $\frac{- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}}{2 x}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

$9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}} \cdot \frac{x}{x} + \frac{- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}}{2 x} \cdot \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 2.1.2.4.3.10

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

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Step 2.1.2.4.3.10.1

Multiply $\frac{9 e^{- x}}{2 x^{\frac{1}{2}}}$ by $\frac{x}{x}$ .

$9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x} x}{2 x^{\frac{1}{2}} x} + \frac{- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}}{2 x} \cdot \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 2.1.2.4.3.10.2

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

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

Step 2.1.2.4.3.10.3

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

$9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x} x}{2 x^{1 + \frac{1}{2}}} + \frac{- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}}{2 x} \cdot \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 2.1.2.4.3.10.4

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

$9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x} x}{2 x^{\frac{2}{2} + \frac{1}{2}}} + \frac{- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}}{2 x} \cdot \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 2.1.2.4.3.10.5

Combine the numerators over the common denominator.

$9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x} x}{2 x^{\frac{2 + 1}{2}}} + \frac{- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}}{2 x} \cdot \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 2.1.2.4.3.10.6

Add $2$ and $1$ .

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

Step 2.1.2.4.3.10.7

Multiply $\frac{- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}}{2 x}$ by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

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

Step 2.1.2.4.3.10.8

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

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Step 2.1.2.4.3.10.8.1

Move $x^{\frac{1}{2}}$ .

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

Step 2.1.2.4.3.10.8.2

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

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Step 2.1.2.4.3.10.8.2.1

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

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

Step 2.1.2.4.3.10.8.2.2

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

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

Step 2.1.2.4.3.10.8.3

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

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

Step 2.1.2.4.3.10.8.4

Combine the numerators over the common denominator.

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

Step 2.1.2.4.3.10.8.5

Add $1$ and $2$ .

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

Step 2.1.2.4.3.11

Combine the numerators over the common denominator.

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

Step 2.1.2.4.4

Reorder terms.

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

Step 2.1.2.4.5

Simplify each term.

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Step 2.1.2.4.5.1

Simplify the numerator.

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Step 2.1.2.4.5.1.1

Factor $x^{\frac{1}{2}}$ out of $- 9 e^{- x} x + (- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}) x^{\frac{1}{2}}$ .

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Step 2.1.2.4.5.1.1.1

Reorder the expression.

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Step 2.1.2.4.5.1.1.1.1

Move $e^{- x}$ .

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

Step 2.1.2.4.5.1.1.1.2

Reorder $- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}$ and $x^{\frac{1}{2}}$ .

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

Step 2.1.2.4.5.1.1.2

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

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

Step 2.1.2.4.5.1.1.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (- 9 x^{\frac{1}{2}} e^{- x}) + x^{\frac{1}{2}} (- 9 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}})$ .

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

Step 2.1.2.4.5.1.2

Subtract $9 x^{\frac{1}{2}} e^{- x}$ from $- 9 x^{\frac{1}{2}} e^{- x}$ .

$9 e^{- x} x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}} (- 18 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}})}{2 x^{\frac{3}{2}}}$

Step 2.1.2.4.5.1.3

Factor $9 e^{- x}$ out of $- 18 x^{\frac{1}{2}} e^{- x} - \frac{9 e^{- x}}{2 x^{\frac{1}{2}}}$ .

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Step 2.1.2.4.5.1.3.1

Factor $9 e^{- x}$ out of $- 18 x^{\frac{1}{2}} e^{- x}$ .

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

Step 2.1.2.4.5.1.3.2

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

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

Step 2.1.2.4.5.1.3.3

Factor $9 e^{- x}$ out of $9 e^{- x} (- 2 x^{\frac{1}{2}}) + 9 e^{- x} (- \frac{1}{2 x^{\frac{1}{2}}})$ .

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

Step 2.1.2.4.5.1.4

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

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Step 2.1.2.4.5.1.4.1

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

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

Step 2.1.2.4.5.1.4.2

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

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

Step 2.1.2.4.5.1.4.3

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

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

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

Step 2.1.2.4.5.1.5

Combine exponents.

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Step 2.1.2.4.5.1.5.1

Factor out negative.

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

Step 2.1.2.4.5.1.5.2

Multiply $9$ by $- 1$ .

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

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

Step 2.1.2.4.5.1.6

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

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

Step 2.1.2.4.5.1.7

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

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

Step 2.1.2.4.5.1.8

Combine the numerators over the common denominator.

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

Step 2.1.2.4.5.1.9

Simplify the numerator.

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Step 2.1.2.4.5.1.9.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.2.4.5.1.9.2

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

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Step 2.1.2.4.5.1.9.2.1

Move $x^{\frac{1}{2}}$ .

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

Step 2.1.2.4.5.1.9.2.2

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

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

Step 2.1.2.4.5.1.9.2.3

Combine the numerators over the common denominator.

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

Step 2.1.2.4.5.1.9.2.4

Add $1$ and $1$ .

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

Step 2.1.2.4.5.1.9.2.5

Divide $2$ by $2$ .

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

Step 2.1.2.4.5.1.9.3

Simplify $2 \cdot 2 x^{1}$ .

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

Step 2.1.2.4.5.1.9.4

Multiply $2$ by $2$ .

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

Step 2.1.2.4.5.1.10

Combine exponents.

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Step 2.1.2.4.5.1.10.1

Combine $x^{\frac{1}{2}}$ and $\frac{4 x + 1}{2 x^{\frac{1}{2}}}$ .

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

Step 2.1.2.4.5.1.10.2

Combine $\frac{x^{\frac{1}{2}} (4 x + 1)}{2 x^{\frac{1}{2}}}$ and $- 9$ .

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

Step 2.1.2.4.5.1.10.3

Combine $\frac{x^{\frac{1}{2}} (4 x + 1) \cdot - 9}{2 x^{\frac{1}{2}}}$ and $e^{- x}$ .

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

Step 2.1.2.4.5.1.11

Reduce the expression $\frac{x^{\frac{1}{2}} (4 x + 1) \cdot - 9 e^{- x}}{2 x^{\frac{1}{2}}}$ by cancelling the common factors.

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Step 2.1.2.4.5.1.11.1

Cancel the common factor.

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

Step 2.1.2.4.5.1.11.2

Rewrite the expression.

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

Step 2.1.2.4.5.1.12

Move $- 9$ to the left of $4 x + 1$ .

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

Step 2.1.2.4.5.1.13

Move the negative in front of the fraction.

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

Step 2.1.2.4.5.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.2.4.5.3

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

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Step 2.1.2.4.5.3.1

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

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

Step 2.1.2.4.5.3.2

Multiply $2$ by $2$ .

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

Step 2.1.2.4.6

To write $9 e^{- x} x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{4 x^{\frac{3}{2}}}{4 x^{\frac{3}{2}}}$ .

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

Step 2.1.2.4.7

Combine $9 e^{- x} x^{\frac{1}{2}}$ and $\frac{4 x^{\frac{3}{2}}}{4 x^{\frac{3}{2}}}$ .

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

Step 2.1.2.4.8

Combine the numerators over the common denominator.

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

Step 2.1.2.4.9

Simplify the numerator.

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Step 2.1.2.4.9.1

Factor $9 e^{- x}$ out of $9 e^{- x} x^{\frac{1}{2}} (4 x^{\frac{3}{2}}) - 9 (4 x + 1) e^{- x}$ .

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Step 2.1.2.4.9.1.1

Factor $9 e^{- x}$ out of $9 e^{- x} x^{\frac{1}{2}} (4 x^{\frac{3}{2}})$ .

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

Step 2.1.2.4.9.1.2

Factor $9 e^{- x}$ out of $- 9 (4 x + 1) e^{- x}$ .

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

Step 2.1.2.4.9.1.3

Factor $9 e^{- x}$ out of $9 e^{- x} (x^{\frac{1}{2}} (4 x^{\frac{3}{2}})) + 9 e^{- x} (- (4 x + 1))$ .

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

Step 2.1.2.4.9.2

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

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Step 2.1.2.4.9.2.1

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

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

Step 2.1.2.4.9.2.2

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

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

Step 2.1.2.4.9.2.3

Combine the numerators over the common denominator.

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

Step 2.1.2.4.9.2.4

Add $3$ and $1$ .

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

Step 2.1.2.4.9.2.5

Divide $4$ by $2$ .

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

Step 2.1.2.4.9.3

Simplify each term.

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Step 2.1.2.4.9.3.1

Move $4$ to the left of $x^{2}$ .

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

Step 2.1.2.4.9.3.2

Apply the distributive property.

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

Step 2.1.2.4.9.3.3

Multiply $4$ by $- 1$ .

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

Step 2.1.2.4.9.3.4

Multiply $- 1$ by $1$ .

$f'' (x) = \frac{9 e^{- x} (4 x^{2} - 4 x - 1)}{4 x^{\frac{3}{2}}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{9 e^{- x} (4 x^{2} - 4 x - 1)}{4 x^{\frac{3}{2}}}$ .

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

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{9 e^{- x} (4 x^{2} - 4 x - 1)}{4 x^{\frac{3}{2}}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{9 e^{- x} (4 x^{2} - 4 x - 1)}{4 x^{\frac{3}{2}}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$9 e^{- x} (4 x^{2} - 4 x - 1) = 0$

Step 2.2.3

Solve the equation for $x$ .

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Step 2.2.3.1

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

$e^{- x} = 0$

$4 x^{2} - 4 x - 1 = 0$

Step 2.2.3.2

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

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Step 2.2.3.2.1

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

$e^{- x} = 0$

Step 2.2.3.2.2

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

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Step 2.2.3.2.2.1

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

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

Step 2.2.3.2.2.2

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

Undefined

Step 2.2.3.2.2.3

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

No solution

Step 2.2.3.3

Set $4 x^{2} - 4 x - 1$ equal to $0$ and solve for $x$ .

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Step 2.2.3.3.1

Set $4 x^{2} - 4 x - 1$ equal to $0$ .

$4 x^{2} - 4 x - 1 = 0$

Step 2.2.3.3.2

Solve $4 x^{2} - 4 x - 1 = 0$ for $x$ .

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Step 2.2.3.3.2.1

Use the quadratic formula to find the solutions.

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

Step 2.2.3.3.2.2

Substitute the values $a = 4$ , $b = - 4$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 2.2.3.3.2.3

Simplify.

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Step 2.2.3.3.2.3.1

Simplify the numerator.

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Step 2.2.3.3.2.3.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 4 \cdot - 1}}{2 \cdot 4}$

Step 2.2.3.3.2.3.1.2

Multiply $- 4 \cdot 4 \cdot - 1$ .

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Step 2.2.3.3.2.3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{4 \pm \sqrt{16 - 16 \cdot - 1}}{2 \cdot 4}$

Step 2.2.3.3.2.3.1.2.2

Multiply $- 16$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 16}}{2 \cdot 4}$

Step 2.2.3.3.2.3.1.3

Add $16$ and $16$ .

$x = \frac{4 \pm \sqrt{32}}{2 \cdot 4}$

Step 2.2.3.3.2.3.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.2.3.3.2.3.1.4.1

Factor $16$ out of $32$ .

$x = \frac{4 \pm \sqrt{16 (2)}}{2 \cdot 4}$

Step 2.2.3.3.2.3.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 4}$

Step 2.2.3.3.2.3.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 4 \sqrt{2}}{2 \cdot 4}$

Step 2.2.3.3.2.3.2

Multiply $2$ by $4$ .

$x = \frac{4 \pm 4 \sqrt{2}}{8}$

Step 2.2.3.3.2.3.3

Simplify $\frac{4 \pm 4 \sqrt{2}}{8}$ .

$x = \frac{1 \pm \sqrt{2}}{2}$

Step 2.2.3.3.2.4

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

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Step 2.2.3.3.2.4.1

Simplify the numerator.

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Step 2.2.3.3.2.4.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 4 \cdot - 1}}{2 \cdot 4}$

Step 2.2.3.3.2.4.1.2

Multiply $- 4 \cdot 4 \cdot - 1$ .

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Step 2.2.3.3.2.4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{4 \pm \sqrt{16 - 16 \cdot - 1}}{2 \cdot 4}$

Step 2.2.3.3.2.4.1.2.2

Multiply $- 16$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 16}}{2 \cdot 4}$

Step 2.2.3.3.2.4.1.3

Add $16$ and $16$ .

$x = \frac{4 \pm \sqrt{32}}{2 \cdot 4}$

Step 2.2.3.3.2.4.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.2.3.3.2.4.1.4.1

Factor $16$ out of $32$ .

$x = \frac{4 \pm \sqrt{16 (2)}}{2 \cdot 4}$

Step 2.2.3.3.2.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 4}$

Step 2.2.3.3.2.4.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 4 \sqrt{2}}{2 \cdot 4}$

Step 2.2.3.3.2.4.2

Multiply $2$ by $4$ .

$x = \frac{4 \pm 4 \sqrt{2}}{8}$

Step 2.2.3.3.2.4.3

Simplify $\frac{4 \pm 4 \sqrt{2}}{8}$ .

$x = \frac{1 \pm \sqrt{2}}{2}$

Step 2.2.3.3.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{2}}{2}$

Step 2.2.3.3.2.5

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

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Step 2.2.3.3.2.5.1

Simplify the numerator.

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Step 2.2.3.3.2.5.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 4 \cdot - 1}}{2 \cdot 4}$

Step 2.2.3.3.2.5.1.2

Multiply $- 4 \cdot 4 \cdot - 1$ .

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Step 2.2.3.3.2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{4 \pm \sqrt{16 - 16 \cdot - 1}}{2 \cdot 4}$

Step 2.2.3.3.2.5.1.2.2

Multiply $- 16$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 16}}{2 \cdot 4}$

Step 2.2.3.3.2.5.1.3

Add $16$ and $16$ .

$x = \frac{4 \pm \sqrt{32}}{2 \cdot 4}$

Step 2.2.3.3.2.5.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.2.3.3.2.5.1.4.1

Factor $16$ out of $32$ .

$x = \frac{4 \pm \sqrt{16 (2)}}{2 \cdot 4}$

Step 2.2.3.3.2.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 4}$

Step 2.2.3.3.2.5.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 4 \sqrt{2}}{2 \cdot 4}$

Step 2.2.3.3.2.5.2

Multiply $2$ by $4$ .

$x = \frac{4 \pm 4 \sqrt{2}}{8}$

Step 2.2.3.3.2.5.3

Simplify $\frac{4 \pm 4 \sqrt{2}}{8}$ .

$x = \frac{1 \pm \sqrt{2}}{2}$

Step 2.2.3.3.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{2}}{2}$

Step 2.2.3.3.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{2}}{2}, \frac{1 - \sqrt{2}}{2}$

Step 2.2.3.4

The final solution is all the values that make $9 e^{- x} (4 x^{2} - 4 x - 1) = 0$ true.

$x = \frac{1 + \sqrt{2}}{2}, \frac{1 - \sqrt{2}}{2}$

Step 2.2.4

Exclude the solutions that do not make $\frac{9 e^{- x} (4 x^{2} - 4 x - 1)}{4 x^{\frac{3}{2}}} = 0$ true.

$x = \frac{1 + \sqrt{2}}{2}$

Step 3

Find the domain of $f (x) = 9 \sqrt{x} e^{- x}$ .

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Step 3.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 3.2

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$[0, \frac{1 + \sqrt{2}}{2}) \cup (\frac{1 + \sqrt{2}}{2}, \infty)$

Step 5

Substitute any number from the interval $[0, \frac{1 + \sqrt{2}}{2})$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (1) = \frac{9 e^{- (1)} (4 {(1)}^{2} - 4 \cdot 1 - 1)}{4 {(1)}^{\frac{3}{2}}}$

Step 5.2

Simplify the result.

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Step 5.2.1

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

$f'' (1) = \frac{9 (4 {(1)}^{2} - 4 - 1)}{4 {(1)}^{\frac{3}{2}} e}$

Step 5.2.2

Simplify the numerator.

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Step 5.2.2.1

One to any power is one.

$f'' (1) = \frac{9 (4 \cdot 1 - 4 - 1)}{4 \cdot (1^{\frac{3}{2}} e)}$

Step 5.2.2.2

Multiply $4$ by $1$ .

$f'' (1) = \frac{9 (4 - 4 - 1)}{4 \cdot (1^{\frac{3}{2}} e)}$

Step 5.2.2.3

Subtract $4$ from $4$ .

$f'' (1) = \frac{9 (0 - 1)}{4 \cdot (1^{\frac{3}{2}} e)}$

Step 5.2.2.4

Subtract $1$ from $0$ .

$f'' (1) = \frac{9 \cdot - 1}{4 \cdot (1^{\frac{3}{2}} e)}$

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

One to any power is one.

$f'' (1) = \frac{9 \cdot - 1}{4 \cdot (1 e)}$

Step 5.2.3.2

Multiply $4$ by $1$ .

$f'' (1) = \frac{9 \cdot - 1}{4 e}$

Step 5.2.4

Simplify the expression.

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Step 5.2.4.1

Multiply $9$ by $- 1$ .

$f'' (1) = \frac{- 9}{4 e}$

Step 5.2.4.2

Move the negative in front of the fraction.

$f'' (1) = - \frac{9}{4 e}$

Step 5.2.5

The final answer is $- \frac{9}{4 e}$ .

$- \frac{9}{4 e}$

Step 5.3

The graph is concave down on the interval $[0, \frac{1 + \sqrt{2}}{2})$ because $f'' (1)$ is negative.

Concave down on $[0, \frac{1 + \sqrt{2}}{2})$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(\frac{1 + \sqrt{2}}{2}, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (4) = \frac{9 e^{- (4)} (4 {(4)}^{2} - 4 \cdot 4 - 1)}{4 {(4)}^{\frac{3}{2}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $4$ by ${(4)}^{2}$ by adding the exponents.

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Step 6.2.1.1

Multiply $4$ by ${(4)}^{2}$ .

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Step 6.2.1.1.1

Raise $4$ to the power of $1$ .

$f'' (4) = \frac{9 e^{- (4)} (4 {(4)}^{2} - 4 \cdot 4 - 1)}{4 {(4)}^{\frac{3}{2}}}$

Step 6.2.1.1.2

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

$f'' (4) = \frac{9 e^{- (4)} (4^{1 + 2} - 4 \cdot 4 - 1)}{4 {(4)}^{\frac{3}{2}}}$

Step 6.2.1.2

Add $1$ and $2$ .

$f'' (4) = \frac{9 e^{- (4)} (4^{3} - 4 \cdot 4 - 1)}{4 {(4)}^{\frac{3}{2}}}$

Step 6.2.2

Multiply $4$ by ${(4)}^{\frac{3}{2}}$ by adding the exponents.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

Raise $4$ to the power of $1$ .

$f'' (4) = \frac{9 e^{- (4)} (4^{3} - 4 \cdot 4 - 1)}{4 {(4)}^{\frac{3}{2}}}$

Step 6.2.2.1.2

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

$f'' (4) = \frac{9 e^{- (4)} (4^{3} - 4 \cdot 4 - 1)}{4^{1 + \frac{3}{2}}}$

Step 6.2.2.2

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

$f'' (4) = \frac{9 e^{- (4)} (4^{3} - 4 \cdot 4 - 1)}{4^{\frac{2}{2} + \frac{3}{2}}}$

Step 6.2.2.3

Combine the numerators over the common denominator.

$f'' (4) = \frac{9 e^{- (4)} (4^{3} - 4 \cdot 4 - 1)}{4^{\frac{2 + 3}{2}}}$

Step 6.2.2.4

Add $2$ and $3$ .

$f'' (4) = \frac{9 e^{- (4)} (4^{3} - 4 \cdot 4 - 1)}{4^{\frac{5}{2}}}$

Step 6.2.3

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

$f'' (4) = \frac{9 (4^{3} - 4 \cdot 4 - 1)}{4^{\frac{5}{2}} e^{4}}$

Step 6.2.4

Simplify the numerator.

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Step 6.2.4.1

Raise $4$ to the power of $3$ .

$f'' (4) = \frac{9 (64 - 4 \cdot 4 - 1)}{4^{\frac{5}{2}} e^{4}}$

Step 6.2.4.2

Multiply $- 4$ by $4$ .

$f'' (4) = \frac{9 (64 - 16 - 1)}{4^{\frac{5}{2}} e^{4}}$

Step 6.2.4.3

Subtract $16$ from $64$ .

$f'' (4) = \frac{9 (48 - 1)}{4^{\frac{5}{2}} e^{4}}$

Step 6.2.4.4

Subtract $1$ from $48$ .

$f'' (4) = \frac{9 \cdot 47}{4^{\frac{5}{2}} e^{4}}$

Step 6.2.5

Simplify the denominator.

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Step 6.2.5.1

Rewrite $4$ as $2^{2}$ .

$f'' (4) = \frac{9 \cdot 47}{{(2^{2})}^{\frac{5}{2}} e^{4}}$

Step 6.2.5.2

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

$f'' (4) = \frac{9 \cdot 47}{2^{2 (\frac{5}{2})} e^{4}}$

Step 6.2.5.3

Cancel the common factor of $2$ .

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Step 6.2.5.3.1

Cancel the common factor.

$f'' (4) = \frac{9 \cdot 47}{2^{2 (\frac{5}{2})} e^{4}}$

Step 6.2.5.3.2

Rewrite the expression.

$f'' (4) = \frac{9 \cdot 47}{2^{5} e^{4}}$

Step 6.2.5.4

Raise $2$ to the power of $5$ .

$f'' (4) = \frac{9 \cdot 47}{32 e^{4}}$

Step 6.2.6

Multiply $9$ by $47$ .

$f'' (4) = \frac{423}{32 e^{4}}$

Step 6.2.7

The final answer is $\frac{423}{32 e^{4}}$ .

$\frac{423}{32 e^{4}}$

Step 6.3

The graph is concave up on the interval $(\frac{1 + \sqrt{2}}{2}, \infty)$ because $f'' (4)$ is positive.

Concave up on $(\frac{1 + \sqrt{2}}{2}, \infty)$ since $f'' (x)$ is positive

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $[0, \frac{1 + \sqrt{2}}{2})$ since $f'' (x)$ is negative

Concave up on $(\frac{1 + \sqrt{2}}{2}, \infty)$ since $f'' (x)$ is positive

Step 8