Calculus Examples

$f (x) = 2 \sqrt{x} {(e)}^{- x}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

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

Step 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 $g (x) = e^{- x}$ .

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

Step 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 $g (x) = - x$ .

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

Differentiate.

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Simplify the expression.

Tap for more steps...

Step 1.4.3.1

Multiply $- 1$ by $1$ .

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

Step 1.4.3.2

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

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

Step 1.4.3.3

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

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

Step 1.4.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Combine the numerators over the common denominator.

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

Step 1.8

Simplify the numerator.

Tap for more steps...

Step 1.8.1

Multiply $- 1$ by $2$ .

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

Step 1.8.2

Subtract $2$ from $1$ .

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

Step 1.9

Move the negative in front of the fraction.

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

Simplify.

Tap for more steps...

Step 1.13.1

Apply the distributive property.

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

Step 1.13.2

Combine terms.

Tap for more steps...

Step 1.13.2.1

Multiply $- 1$ by $2$ .

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

Step 1.13.2.2

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

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

Step 1.13.2.3

Cancel the common factor.

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

Step 1.13.2.4

Rewrite the expression.

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

Step 1.13.3

Reorder terms.

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

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

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

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

Step 2.2.3

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

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

Step 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 $g (x) = - x$ .

Tap for more steps...

Step 2.2.4.1

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

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

Step 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 $a$ = $e$ .

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

Step 2.2.4.3

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.2.9

Combine the numerators over the common denominator.

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

Step 2.2.10

Simplify the numerator.

Tap for more steps...

Step 2.2.10.1

Multiply $- 1$ by $2$ .

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

Step 2.2.10.2

Subtract $2$ from $1$ .

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

Step 2.2.11

Move the negative in front of the fraction.

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

Step 2.2.12

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

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

Step 2.2.13

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

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

Step 2.2.14

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

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

Step 2.2.15

Multiply $- 1$ by $1$ .

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

Step 2.2.16

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

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

Step 2.2.17

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

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

Tap for more steps...

Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Multiply $- 1$ by $1$ .

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

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

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

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

Step 2.3.11

Combine the numerators over the common denominator.

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

Step 2.3.12

Simplify the numerator.

Tap for more steps...

Step 2.3.12.1

Multiply $- 1$ by $2$ .

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

Step 2.3.12.2

Subtract $2$ from $1$ .

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

Step 2.3.13

Move the negative in front of the fraction.

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

Step 2.3.14

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

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

Step 2.3.15

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

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

Step 2.3.16

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

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

Step 2.3.17

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

Tap for more steps...

Step 2.3.17.1

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

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

Step 2.3.17.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.17.2.1

Cancel the common factor.

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

Step 2.3.17.2.2

Rewrite the expression.

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

Step 2.3.18

Simplify.

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

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

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

Step 2.4.2.2

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

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

Step 2.4.2.3

Cancel the common factors.

Tap for more steps...

Step 2.4.2.3.1

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

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

Step 2.4.2.3.2

Cancel the common factor.

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

Step 2.4.2.3.3

Rewrite the expression.

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

Step 2.4.2.4

Move the negative in front of the fraction.

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

Step 2.4.2.5

Multiply $- 1$ by $- 2$ .

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

Step 2.4.2.6

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

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

Step 2.4.2.7

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

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

Step 2.4.2.8

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

Tap for more steps...

Step 2.4.2.8.1

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

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

Step 2.4.2.8.2

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

Tap for more steps...

Step 2.4.2.8.2.1

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

Tap for more steps...

Step 2.4.2.8.2.1.1

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

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

Step 2.4.2.8.2.1.2

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

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

Step 2.4.2.8.2.2

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

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

Step 2.4.2.8.2.3

Combine the numerators over the common denominator.

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

Step 2.4.2.8.2.4

Add $1$ and $2$ .

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

Step 2.4.2.8.3

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

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

Step 2.4.2.8.4

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

Tap for more steps...

Step 2.4.2.8.4.1

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

Tap for more steps...

Step 2.4.2.8.4.1.1

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

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

Step 2.4.2.8.4.1.2

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

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

Step 2.4.2.8.4.2

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

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

Step 2.4.2.8.4.3

Combine the numerators over the common denominator.

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

Step 2.4.2.8.4.4

Add $2$ and $1$ .

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

Step 2.4.2.9

Combine the numerators over the common denominator.

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

Step 2.4.2.10

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

Tap for more steps...

Step 2.4.2.10.1

Reorder the expression.

Tap for more steps...

Step 2.4.2.10.1.1

Move $e^{- x}$ .

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

Step 2.4.2.10.1.2

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

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

Step 2.4.2.10.1.3

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

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

Step 2.4.2.10.2

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

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

Step 2.4.2.10.3

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

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

Step 2.4.2.11

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

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

Step 2.4.2.12

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

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

Step 2.4.2.13

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

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

Step 2.4.2.14

Simplify the denominator.

Tap for more steps...

Step 2.4.2.14.1

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

Tap for more steps...

Step 2.4.2.14.1.1

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

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

Step 2.4.2.14.1.2

Combine the numerators over the common denominator.

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

Step 2.4.2.14.1.3

Subtract $1$ from $3$ .

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

Step 2.4.2.14.1.4

Divide $2$ by $2$ .

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

Step 2.4.2.14.2

Simplify $x^{1}$ .

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

Step 2.4.2.15

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

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

Step 2.4.2.16

Combine the numerators over the common denominator.

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

Step 2.4.2.17

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

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

Step 2.4.2.18

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

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

Step 2.4.2.19

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

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

Step 2.4.2.20

Combine the numerators over the common denominator.

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

Step 2.4.2.21

Add $2$ and $1$ .

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

Step 2.4.3

Reorder terms.

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

Step 2.4.4

Simplify the numerator.

Tap for more steps...

Step 2.4.4.1

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

Tap for more steps...

Step 2.4.4.1.1

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

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

Step 2.4.4.1.2

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

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

Step 2.4.4.1.3

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

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

Step 2.4.4.1.4

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

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

Step 2.4.4.1.5

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

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

Step 2.4.4.2

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

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

Step 2.4.4.3

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

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

Step 2.4.4.4

Combine the numerators over the common denominator.

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

Step 2.4.4.5

Simplify the numerator.

Tap for more steps...

Step 2.4.4.5.1

Multiply $2$ by $2$ .

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

Step 2.4.4.5.2

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

Tap for more steps...

Step 2.4.4.5.2.1

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

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

Step 2.4.4.5.2.2

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

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

Step 2.4.4.5.2.3

Combine the numerators over the common denominator.

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

Step 2.4.4.5.2.4

Add $1$ and $3$ .

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

Step 2.4.4.5.2.5

Divide $4$ by $2$ .

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

Step 2.4.4.5.3

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 2.4.4.5.4

Rewrite $1$ as $1^{2}$ .

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

Step 2.4.4.5.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

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

Step 2.4.4.6

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

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

Step 2.4.4.7

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

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

Step 2.4.4.8

Combine the numerators over the common denominator.

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

Step 2.4.4.9

Simplify the numerator.

Tap for more steps...

Step 2.4.4.9.1

Rewrite using the commutative property of multiplication.

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

Step 2.4.4.9.2

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

Tap for more steps...

Step 2.4.4.9.2.1

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

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

Step 2.4.4.9.2.2

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

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

Step 2.4.4.9.2.3

Combine the numerators over the common denominator.

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

Step 2.4.4.9.2.4

Add $1$ and $1$ .

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

Step 2.4.4.9.2.5

Divide $2$ by $2$ .

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

Step 2.4.4.9.3

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

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

Step 2.4.4.9.4

Multiply $- 2$ by $2$ .

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

Step 2.4.4.9.5

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

Tap for more steps...

Step 2.4.4.9.5.1

Apply the distributive property.

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

Step 2.4.4.9.5.2

Apply the distributive property.

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

Step 2.4.4.9.5.3

Apply the distributive property.

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

Step 2.4.4.9.6

Combine the opposite terms in $2 x (2 x) + 2 x \cdot - 1 + 1 (2 x) + 1 \cdot - 1$ .

Tap for more steps...

Step 2.4.4.9.6.1

Reorder the factors in the terms $2 x \cdot - 1$ and $1 (2 x)$ .

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

Step 2.4.4.9.6.2

Add $- 1 \cdot 2 x$ and $1 \cdot 2 x$ .

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

Step 2.4.4.9.6.3

Add $2 x (2 x)$ and $0$ .

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

Step 2.4.4.9.7

Simplify each term.

Tap for more steps...

Step 2.4.4.9.7.1

Rewrite using the commutative property of multiplication.

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

Step 2.4.4.9.7.2

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

Tap for more steps...

Step 2.4.4.9.7.2.1

Move $x$ .

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

Step 2.4.4.9.7.2.2

Multiply $x$ by $x$ .

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

Step 2.4.4.9.7.3

Multiply $2$ by $2$ .

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

Step 2.4.4.9.7.4

Multiply $- 1$ by $1$ .

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

Step 2.4.4.9.8

Reorder terms.

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

Step 2.4.5

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

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

Step 2.4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.7

Combine.

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

Step 2.4.8

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

Tap for more steps...

Step 2.4.8.1

Move $x$ .

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

Step 2.4.8.2

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

Tap for more steps...

Step 2.4.8.2.1

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

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

Step 2.4.8.2.2

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

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

Step 2.4.8.3

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

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

Step 2.4.8.4

Combine the numerators over the common denominator.

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

Step 2.4.8.5

Add $2$ and $1$ .

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

Step 2.4.9

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

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

Step 3

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

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

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.2

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

Step 4.1.3

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 $g (x) = - x$ .

Tap for more steps...

Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Differentiate.

Tap for more steps...

Step 4.1.4.1

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

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

Step 4.1.4.2

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

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

Step 4.1.4.3

Simplify the expression.

Tap for more steps...

Step 4.1.4.3.1

Multiply $- 1$ by $1$ .

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

Step 4.1.4.3.2

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

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

Step 4.1.4.3.3

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

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

Step 4.1.4.4

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

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

Step 4.1.5

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

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

Step 4.1.6

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

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

Step 4.1.7

Combine the numerators over the common denominator.

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

Step 4.1.8

Simplify the numerator.

Tap for more steps...

Step 4.1.8.1

Multiply $- 1$ by $2$ .

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

Step 4.1.8.2

Subtract $2$ from $1$ .

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

Step 4.1.9

Move the negative in front of the fraction.

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

Step 4.1.10

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

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

Step 4.1.11

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

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

Step 4.1.12

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

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

Step 4.1.13

Simplify.

Tap for more steps...

Step 4.1.13.1

Apply the distributive property.

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

Step 4.1.13.2

Combine terms.

Tap for more steps...

Step 4.1.13.2.1

Multiply $- 1$ by $2$ .

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

Step 4.1.13.2.2

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

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

Step 4.1.13.2.3

Cancel the common factor.

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

Step 4.1.13.2.4

Rewrite the expression.

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

Step 4.1.13.3

Reorder terms.

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

Step 4.2

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

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

Step 5

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

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Find a common factor $e^{- 1 x}$ that is present in each term.

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

Step 5.3

Substitute $u$ for $e^{- 1 x}$ .

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

Step 5.4

Solve for $u$ .

Tap for more steps...

Step 5.4.1

Move $\frac{e^{- x}}{x^{\frac{1}{2}}}$ to the right side of the equation by subtracting it from both sides.

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

Step 5.4.2

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

Tap for more steps...

Step 5.4.2.1

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

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

Step 5.4.2.2

Apply the product rule to $e x^{- x}$ .

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

Step 5.4.2.3

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

Tap for more steps...

Step 5.4.2.3.1

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

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

Step 5.4.2.3.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.4.2.3.2.1

Factor $x$ out of $- x$ .

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

Step 5.4.2.3.2.2

Factor $x$ out of $2 x$ .

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

Step 5.4.2.3.2.3

Cancel the common factor.

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

Step 5.4.2.3.2.4

Rewrite the expression.

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

$- 2 e^{\frac{2 x - 1}{2 x}} x^{- \frac{2 x - 1}{2}} = - \frac{e^{- x}}{x^{\frac{1}{2}}}$

Step 5.4.3

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 5.4.3.1

Add $\frac{e^{- x}}{x^{\frac{1}{2}}}$ to both sides of the equation.

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

Step 5.4.3.2

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

$- 2 e^{\frac{2 x - 1}{2 x}} x^{- \frac{2 x - 1}{2}} \cdot \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}} + \frac{e^{- x}}{x^{\frac{1}{2}}} = 0$

Step 5.4.3.3

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

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

Step 5.4.3.4

Combine the numerators over the common denominator.

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

Step 5.4.3.5

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

Tap for more steps...

Step 5.4.3.5.1

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

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

Step 5.4.3.5.2

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

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

Step 5.4.3.5.3

Combine the numerators over the common denominator.

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

Step 5.4.3.5.4

Simplify each term.

Tap for more steps...

Step 5.4.3.5.4.1

Apply the distributive property.

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

Step 5.4.3.5.4.2

Multiply $2$ by $- 1$ .

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

Step 5.4.3.5.4.3

Multiply $- 1$ by $- 1$ .

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

Step 5.4.3.5.5

Add $1$ and $1$ .

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

Step 5.4.3.5.6

Cancel the common factor of $- 2 x + 2$ and $2$ .

Tap for more steps...

Step 5.4.3.5.6.1

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

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

Step 5.4.3.5.6.2

Factor $2$ out of $2$ .

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

Step 5.4.3.5.6.3

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

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

Step 5.4.3.5.6.4

Cancel the common factors.

Tap for more steps...

Step 5.4.3.5.6.4.1

Factor $2$ out of $2$ .

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

Step 5.4.3.5.6.4.2

Cancel the common factor.

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

Step 5.4.3.5.6.4.3

Rewrite the expression.

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

Step 5.4.3.5.6.4.4

Divide $- x + 1$ by $1$ .

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

Step 5.4.3.6

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

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

Step 5.4.3.7

Factor $- 1$ out of $e^{- x}$ .

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

Step 5.4.3.8

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

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

Step 5.4.3.9

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

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

Step 5.4.3.10

Move the negative in front of the fraction.

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

Step 5.5

Substitute $x$ for $u$ .

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

Step 5.6

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

Tap for more steps...

Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

Step 5.7

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

$e^{- x} = 0$

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

Step 5.8

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

Tap for more steps...

Step 5.8.1

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

$e^{- x} = 0$

Step 5.8.2

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

Tap for more steps...

Step 5.8.2.1

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

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

Step 5.8.2.2

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

Undefined

Step 5.8.2.3

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

No solution

Step 5.9

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

Tap for more steps...

Step 5.9.1

Set $- 2 x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{2}}}$ equal to $0$ .

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

Step 5.9.2

Solve $- 2 x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{2}}} = 0$ for $x$ .

Tap for more steps...

Step 5.9.2.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 5.9.2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{\frac{1}{2}}, 1$

Step 5.9.2.1.2

The LCM of one and any expression is the expression.

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

Step 5.9.2.2

Multiply each term in $- 2 x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ to eliminate the fractions.

Tap for more steps...

Step 5.9.2.2.1

Multiply each term in $- 2 x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{2}}} = 0$ by $x^{\frac{1}{2}}$ .

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

Step 5.9.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.9.2.2.2.1

Simplify each term.

Tap for more steps...

Step 5.9.2.2.2.1.1

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

Tap for more steps...

Step 5.9.2.2.2.1.1.1

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

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

Step 5.9.2.2.2.1.1.2

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

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

Step 5.9.2.2.2.1.1.3

Combine the numerators over the common denominator.

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

Step 5.9.2.2.2.1.1.4

Add $1$ and $1$ .

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

Step 5.9.2.2.2.1.1.5

Divide $2$ by $2$ .

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

Step 5.9.2.2.2.1.2

Simplify $- 2 x^{1}$ .

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

Step 5.9.2.2.2.1.3

Cancel the common factor of $x^{\frac{1}{2}}$ .

Tap for more steps...

Step 5.9.2.2.2.1.3.1

Cancel the common factor.

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

Step 5.9.2.2.2.1.3.2

Rewrite the expression.

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

Step 5.9.2.2.3

Simplify the right side.

Tap for more steps...

Step 5.9.2.2.3.1

Multiply $0$ by $x^{\frac{1}{2}}$ .

$- 2 x + 1 = 0$

Step 5.9.2.3

Solve the equation.

Tap for more steps...

Step 5.9.2.3.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 5.9.2.3.2

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

Tap for more steps...

Step 5.9.2.3.2.1

Divide each term in $- 2 x = - 1$ by $- 2$ .

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

Step 5.9.2.3.2.2

Simplify the left side.

Tap for more steps...

Step 5.9.2.3.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 5.9.2.3.2.2.1.1

Cancel the common factor.

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

Step 5.9.2.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.9.2.3.2.3

Simplify the right side.

Tap for more steps...

Step 5.9.2.3.2.3.1

Dividing two negative values results in a positive value.

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

Step 5.10

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

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

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

Anything raised to $1$ is the base itself.

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

Step 6.1.4

Anything raised to $1$ is the base itself.

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

Step 6.2

Set the denominator in $\frac{e^{- x}}{\sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x} = 0$

Step 6.3

Solve for $x$ .

Tap for more steps...

Step 6.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 6.3.2

Simplify each side of the equation.

Tap for more steps...

Step 6.3.2.1

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

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

Step 6.3.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.2.1

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

Tap for more steps...

Step 6.3.2.2.1.1

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

Tap for more steps...

Step 6.3.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.3.2.2.1.1.2.2

Rewrite the expression.

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

Step 6.3.2.2.1.2

Simplify.

$x = 0^{2}$

Step 6.3.2.3

Simplify the right side.

Tap for more steps...

Step 6.3.2.3.1

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

$x = 0$

Step 6.4

Set the radicand in $\sqrt{x}$ less than $0$ to find where the expression is undefined.

$x < 0$

Step 6.5

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

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 7

Critical points to evaluate.

$x = \frac{1}{2}, 0$

Step 8

Evaluate the second derivative at $x = \frac{1}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

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

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

Step 9.2

Simplify the denominator.

Tap for more steps...

Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.2.3

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

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

Step 9.2.4

Multiply $- 1$ by $1$ .

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

Step 9.2.5

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

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

Step 9.2.6

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

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

Step 9.2.7

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

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

Step 9.2.8

Combine the numerators over the common denominator.

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

Step 9.2.9

Subtract $3$ from $2$ .

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

Step 9.3

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

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

Step 9.4

Simplify the numerator.

Tap for more steps...

Step 9.4.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 9.4.2

One to any power is one.

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

Step 9.4.3

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

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

Step 9.4.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 9.4.4.1

Cancel the common factor.

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

Step 9.4.4.2

Rewrite the expression.

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

Step 9.4.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.4.5.1

Factor $2$ out of $- 4$ .

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

Step 9.4.5.2

Cancel the common factor.

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

Step 9.4.5.3

Rewrite the expression.

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

Step 9.4.6

Subtract $2$ from $1$ .

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

Step 9.4.7

Subtract $1$ from $- 1$ .

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

Step 9.4.8

Move the negative in front of the fraction.

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

Step 9.4.9

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

Tap for more steps...

Step 9.4.9.1

Multiply $- 1$ by $- 1$ .

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

Step 9.4.9.2

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

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

Step 9.5

Simplify the numerator.

Tap for more steps...

Step 9.5.1

Factor out negative.

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

Step 9.5.2

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

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

Step 9.5.3

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

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

Step 9.5.4

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

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

Step 9.5.5

Combine the numerators over the common denominator.

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

Step 9.5.6

Add $2$ and $1$ .

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

Step 9.6

Move the negative in front of the fraction.

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

Step 10

$x = \frac{1}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{1}{2}$ is a local maximum

Step 11

Find the y-value when $x = \frac{1}{2}$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

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

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Remove parentheses.

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

Step 11.2.2

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 11.2.3

Any root of $1$ is $1$ .

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

Step 11.2.4

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 11.2.5

Combine and simplify the denominator.

Tap for more steps...

Step 11.2.5.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 11.2.5.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 11.2.5.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 11.2.5.4

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

$f (\frac{1}{2}) = 2 (\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}) \cdot {(e)}^{- (\frac{1}{2})}$

Step 11.2.5.5

Add $1$ and $1$ .

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

Step 11.2.5.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 11.2.5.6.1

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

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

Step 11.2.5.6.2

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

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

Step 11.2.5.6.3

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

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

Step 11.2.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.5.6.4.1

Cancel the common factor.

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

Step 11.2.5.6.4.2

Rewrite the expression.

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

Step 11.2.5.6.5

Evaluate the exponent.

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

Step 11.2.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.6.1

Cancel the common factor.

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

Step 11.2.6.2

Rewrite the expression.

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

Step 11.2.7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f (\frac{1}{2}) = \sqrt{2} (\frac{1}{e^{\frac{1}{2}}})$

Step 11.2.8

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

$f (\frac{1}{2}) = \frac{\sqrt{2}}{e^{\frac{1}{2}}}$

Step 11.2.9

The final answer is $\frac{\sqrt{2}}{e^{\frac{1}{2}}}$ .

$y = \frac{\sqrt{2}}{e^{\frac{1}{2}}}$

Step 12

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

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

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify the expression.

Tap for more steps...

Step 13.1.1

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

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

Step 13.1.2

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

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

Step 13.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.2.1

Cancel the common factor.

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

Step 13.2.2

Rewrite the expression.

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

Step 13.3

Simplify the expression.

Tap for more steps...

Step 13.3.1

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

$\frac{e^{- (0)} (4 {(0)}^{2} - 4 \cdot 0 - 1)}{2 \cdot 0}$

Step 13.3.2

Multiply $2$ by $0$ .

$\frac{e^{- (0)} (4 {(0)}^{2} - 4 \cdot 0 - 1)}{0}$

Step 13.3.3

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

Undefined

$\frac{e^{- (0)} (4 {(0)}^{2} - 4 \cdot 0 - 1)}{0}$

Step 13.4

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

Undefined

Step 14

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 15