Calculus Examples

$f (x) = \frac{1 - \sqrt{x}}{1 + \sqrt{x}}$

Step 1

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

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Apply basic rules of exponents.

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Step 1.1.1.1.1

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

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

Step 1.1.1.1.2

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

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

Step 1.1.1.2

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

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

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

Add $0$ and $\frac{d}{d x} [- x^{\frac{1}{2}}]$ .

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

Step 1.1.1.3.4

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

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

Step 1.1.1.3.5

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

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

Step 1.1.1.4

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

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

Step 1.1.1.5

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

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

Step 1.1.1.6

Combine the numerators over the common denominator.

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

Step 1.1.1.7

Simplify the numerator.

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Step 1.1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.7.2

Subtract $2$ from $1$ .

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

Step 1.1.1.8

Combine fractions.

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Step 1.1.1.8.1

Move the negative in front of the fraction.

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

Step 1.1.1.8.2

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

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

Step 1.1.1.8.3

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

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

Step 1.1.1.9

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

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

Step 1.1.1.10

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

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

Step 1.1.1.11

Add $0$ and $\frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 1.1.1.12

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

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

Step 1.1.1.13

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

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

Step 1.1.1.14

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

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

Step 1.1.1.15

Combine the numerators over the common denominator.

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

Step 1.1.1.16

Simplify the numerator.

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Step 1.1.1.16.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.16.2

Subtract $2$ from $1$ .

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

Step 1.1.1.17

Move the negative in front of the fraction.

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

Step 1.1.1.18

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

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

Step 1.1.1.19

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

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

Step 1.1.1.20

Simplify.

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Step 1.1.1.20.1

Apply the distributive property.

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

Step 1.1.1.20.2

Apply the distributive property.

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

Step 1.1.1.20.3

Apply the distributive property.

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

Step 1.1.1.20.4

Simplify the numerator.

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Step 1.1.1.20.4.1

Simplify each term.

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Step 1.1.1.20.4.1.1

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

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

Step 1.1.1.20.4.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.1.1.20.4.1.3

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

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Step 1.1.1.20.4.1.3.1

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

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

Step 1.1.1.20.4.1.3.2

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

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

Step 1.1.1.20.4.1.3.3

Cancel the common factor.

Step 1.1.1.20.4.1.3.4

Rewrite the expression.

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

Step 1.1.1.20.4.1.4

Multiply $- 1$ by $1$ .

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

Step 1.1.1.20.4.1.5

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

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

Step 1.1.1.20.4.1.6

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

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Step 1.1.1.20.4.1.6.1

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

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

Step 1.1.1.20.4.1.6.2

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

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

Step 1.1.1.20.4.1.6.3

Cancel the common factor.

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

Step 1.1.1.20.4.1.6.4

Rewrite the expression.

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

Step 1.1.1.20.4.1.7

Multiply $- 1$ by $- 1$ .

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

Step 1.1.1.20.4.1.8

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

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

Step 1.1.1.20.4.2

Combine the opposite terms in $- \frac{1}{2 x^{\frac{1}{2}}} - \frac{1}{2} - \frac{1}{2 x^{\frac{1}{2}}} + \frac{1}{2}$ .

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Step 1.1.1.20.4.2.1

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

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

Step 1.1.1.20.4.2.2

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

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

Step 1.1.1.20.4.3

Combine the numerators over the common denominator.

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

Step 1.1.1.20.4.4

Subtract $1$ from $- 1$ .

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

Step 1.1.1.20.4.5

Factor $2$ out of $- 2$ .

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

Step 1.1.1.20.4.6

Cancel the common factors.

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Step 1.1.1.20.4.6.1

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

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

Step 1.1.1.20.4.6.2

Cancel the common factor.

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

Step 1.1.1.20.4.6.3

Rewrite the expression.

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

Step 1.1.1.20.4.7

Move the negative in front of the fraction.

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

Step 1.1.1.20.5

Combine terms.

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Step 1.1.1.20.5.1

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

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

Step 1.1.1.20.5.2

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

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

Step 1.1.1.20.6

Reorder factors in $- \frac{1}{{(1 + x^{\frac{1}{2}})}^{2} x^{\frac{1}{2}}}$ .

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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Step 1.1.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{1}{2}} {(1 + x^{\frac{1}{2}})}^{2}$ .

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

Replace all occurrences of $u_{1}$ with $x^{\frac{1}{2}} {(1 + x^{\frac{1}{2}})}^{2}$ .

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

Step 1.1.2.4

Multiply.

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Step 1.1.2.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.2.4.2

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 1 + x^{\frac{1}{2}}$ .

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Step 1.1.2.6.1

To apply the Chain Rule, set $u_{2}$ as $1 + x^{\frac{1}{2}}$ .

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

Step 1.1.2.6.2

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

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

Step 1.1.2.6.3

Replace all occurrences of $u_{2}$ with $1 + x^{\frac{1}{2}}$ .

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

Step 1.1.2.7

Differentiate.

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Step 1.1.2.7.1

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

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

Step 1.1.2.7.2

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

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

Step 1.1.2.7.3

Add $0$ and $\frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 1.1.2.7.4

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

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

Step 1.1.2.8

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

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

Step 1.1.2.9

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

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

Step 1.1.2.10

Combine the numerators over the common denominator.

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

Step 1.1.2.11

Simplify the numerator.

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Step 1.1.2.11.1

Multiply $- 1$ by $2$ .

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

Step 1.1.2.11.2

Subtract $2$ from $1$ .

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

Step 1.1.2.12

Simplify terms.

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Step 1.1.2.12.1

Move the negative in front of the fraction.

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

Step 1.1.2.12.2

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

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

Step 1.1.2.12.3

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

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

Step 1.1.2.12.4

Simplify the expression.

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Step 1.1.2.12.4.1

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

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

Step 1.1.2.12.4.2

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

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

Step 1.1.2.12.5

Cancel the common factor.

Step 1.1.2.12.6

Rewrite the expression.

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

Step 1.1.2.12.7

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

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

Step 1.1.2.12.8

Cancel the common factor.

Step 1.1.2.12.9

Simplify.

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Step 1.1.2.12.9.1

Rewrite the expression.

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

Step 1.1.2.12.9.2

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

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

Step 1.1.2.13

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

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

Step 1.1.2.14

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

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

Step 1.1.2.15

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

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

Step 1.1.2.16

Combine the numerators over the common denominator.

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

Step 1.1.2.17

Simplify the numerator.

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Step 1.1.2.17.1

Multiply $- 1$ by $2$ .

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

Step 1.1.2.17.2

Subtract $2$ from $1$ .

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

Step 1.1.2.18

Simplify terms.

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Step 1.1.2.18.1

Move the negative in front of the fraction.

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

Step 1.1.2.18.2

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

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

Step 1.1.2.18.3

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

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

Step 1.1.2.18.4

Simplify.

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Step 1.1.2.18.4.1

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

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

Step 1.1.2.18.4.2

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

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

Step 1.1.2.18.5

Combine the numerators over the common denominator.

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

Step 1.1.2.19

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

Step 1.1.2.20

Simplify the expression.

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Step 1.1.2.20.1

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

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

Step 1.1.2.20.2

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

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

Step 1.1.2.21

Simplify.

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Step 1.1.2.21.1

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

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

Step 1.1.2.21.2

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

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

Step 1.1.2.21.3

Combine terms.

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Step 1.1.2.21.3.1

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

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Step 1.1.2.21.3.1.1

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

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

Step 1.1.2.21.3.1.2

Cancel the common factor of $2$ .

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Step 1.1.2.21.3.1.2.1

Cancel the common factor.

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

Step 1.1.2.21.3.1.2.2

Rewrite the expression.

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

Step 1.1.2.21.3.2

Simplify.

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

Step 1.1.2.21.3.3

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

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Step 1.1.2.21.3.3.1

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

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

Step 1.1.2.21.3.3.2

Multiply $2$ by $2$ .

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

Step 1.1.2.21.4

Reorder the factors of $\frac{1}{x {(1 + x^{\frac{1}{2}})}^{4}} (x^{\frac{1}{2}} + \frac{2 x^{\frac{1}{2}} + {(1 + x^{\frac{1}{2}})}^{2}}{2 x^{\frac{1}{2}}})$ .

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

Step 1.1.2.21.5

Simplify the numerator.

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Step 1.1.2.21.5.1

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

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

Step 1.1.2.21.5.2

Expand $(1 + x^{\frac{1}{2}}) (1 + x^{\frac{1}{2}})$ using the FOIL Method.

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Step 1.1.2.21.5.2.1

Apply the distributive property.

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

Step 1.1.2.21.5.2.2

Apply the distributive property.

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

Step 1.1.2.21.5.2.3

Apply the distributive property.

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

Step 1.1.2.21.5.3

Simplify and combine like terms.

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Step 1.1.2.21.5.3.1

Simplify each term.

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Step 1.1.2.21.5.3.1.1

Multiply $1$ by $1$ .

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

Step 1.1.2.21.5.3.1.2

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

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

Step 1.1.2.21.5.3.1.3

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

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

Step 1.1.2.21.5.3.1.4

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

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Step 1.1.2.21.5.3.1.4.1

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

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

Step 1.1.2.21.5.3.1.4.2

Combine the numerators over the common denominator.

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

Step 1.1.2.21.5.3.1.4.3

Add $1$ and $1$ .

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

Step 1.1.2.21.5.3.1.4.4

Divide $2$ by $2$ .

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

Step 1.1.2.21.5.3.1.5

Simplify $x^{1}$ .

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

Step 1.1.2.21.5.3.2

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

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

Step 1.1.2.21.5.4

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

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

Step 1.1.2.21.5.5

Reorder terms.

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

Step 1.1.2.21.6

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

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

Step 1.1.2.21.7

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

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

Step 1.1.2.21.8

Combine the numerators over the common denominator.

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

Step 1.1.2.21.9

Simplify the numerator.

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Step 1.1.2.21.9.1

Rewrite using the commutative property of multiplication.

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

Step 1.1.2.21.9.2

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

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Step 1.1.2.21.9.2.1

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

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

Step 1.1.2.21.9.2.2

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

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

Step 1.1.2.21.9.2.3

Combine the numerators over the common denominator.

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

Step 1.1.2.21.9.2.4

Add $1$ and $1$ .

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

Step 1.1.2.21.9.2.5

Divide $2$ by $2$ .

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

Step 1.1.2.21.9.3

Simplify $2 x^{1}$ .

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

Step 1.1.2.21.9.4

Add $2 x$ and $x$ .

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

Step 1.1.2.21.9.5

Rewrite $3 x + 4 x^{\frac{1}{2}} + 1$ in a factored form.

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Step 1.1.2.21.9.5.1

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

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

Step 1.1.2.21.9.5.2

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

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

Step 1.1.2.21.9.5.3

Factor by grouping.

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Step 1.1.2.21.9.5.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 1 = 3$ and whose sum is $b = 4$ .

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Step 1.1.2.21.9.5.3.1.1

Factor $4$ out of $4 u$ .

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

Step 1.1.2.21.9.5.3.1.2

Rewrite $4$ as $1$ plus $3$

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

Step 1.1.2.21.9.5.3.1.3

Apply the distributive property.

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

Step 1.1.2.21.9.5.3.1.4

Multiply $u$ by $1$ .

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

Step 1.1.2.21.9.5.3.2

Factor out the greatest common factor from each group.

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Step 1.1.2.21.9.5.3.2.1

Group the first two terms and the last two terms.

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

Step 1.1.2.21.9.5.3.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 1.1.2.21.9.5.3.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

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

Step 1.1.2.21.9.5.4

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

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

Step 1.1.2.21.10

Combine.

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

Step 1.1.2.21.11

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

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Step 1.1.2.21.11.1

Move $x$ .

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

Step 1.1.2.21.11.2

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

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Step 1.1.2.21.11.2.1

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

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

Step 1.1.2.21.11.2.2

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

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

Step 1.1.2.21.11.3

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

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

Step 1.1.2.21.11.4

Combine the numerators over the common denominator.

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

Step 1.1.2.21.11.5

Add $2$ and $1$ .

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

Step 1.1.2.21.12

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

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

Step 1.1.2.21.13

Reorder terms.

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

Step 1.1.2.21.14

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

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

Step 1.1.2.21.15

Cancel the common factors.

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Step 1.1.2.21.15.1

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

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

Step 1.1.2.21.15.2

Cancel the common factor.

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

Step 1.1.2.21.15.3

Rewrite the expression.

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

Step 1.1.3

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

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

Step 1.2

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

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Step 1.2.1

Set the second derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

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

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

Subtract $1$ from both sides of the equation.

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

Step 1.2.3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 1.2.3.3

Simplify the exponent.

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Step 1.2.3.3.1

Simplify the left side.

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Step 1.2.3.3.1.1

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

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Step 1.2.3.3.1.1.1

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

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

Step 1.2.3.3.1.1.2

Raise $3$ to the power of $2$ .

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

Step 1.2.3.3.1.1.3

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

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Step 1.2.3.3.1.1.3.1

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

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

Step 1.2.3.3.1.1.3.2

Cancel the common factor of $2$ .

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Step 1.2.3.3.1.1.3.2.1

Cancel the common factor.

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

Step 1.2.3.3.1.1.3.2.2

Rewrite the expression.

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

Step 1.2.3.3.1.1.4

Simplify.

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

Step 1.2.3.3.2

Simplify the right side.

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Step 1.2.3.3.2.1

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

$9 x = 1$

Step 1.2.3.4

Divide each term in $9 x = 1$ by $9$ and simplify.

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Step 1.2.3.4.1

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Step 1.2.3.4.2

Simplify the left side.

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Step 1.2.3.4.2.1

Cancel the common factor of $9$ .

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Step 1.2.3.4.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Step 1.2.3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Step 1.2.4

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

$No solution$

Step 2

Find the domain of $f (x) = \frac{1 - \sqrt{x}}{1 + \sqrt{x}}$ .

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Step 2.1

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

$x \geq 0$

Step 2.2

Set the denominator in $\frac{1 - \sqrt{x}}{1 + \sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$1 + \sqrt{x} = 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Subtract $1$ from both sides of the equation.

$\sqrt{x} = - 1$

Step 2.3.2

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

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

Step 2.3.3

Simplify each side of the equation.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

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

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Step 2.3.3.2.1.1

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

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Step 2.3.3.2.1.1.1

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

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

Step 2.3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.1.1.2.1

Cancel the common factor.

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

Step 2.3.3.2.1.1.2.2

Rewrite the expression.

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

Step 2.3.3.2.1.2

Simplify.

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

Step 2.3.3.3

Simplify the right side.

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Step 2.3.3.3.1

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

$x = 1$

Step 2.3.4

Exclude the solutions that do not make $1 + \sqrt{x} = 0$ true.

$No solution$

Step 2.4

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 3

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

$[0, \infty)$

Step 4

Substitute any number from the interval $[0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Combine the numerators over the common denominator.

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

Step 4.2.1.4

Add $2$ and $3$ .

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

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

Step 4.2.2

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

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

Step 4.3

The graph is concave up on the interval $[0, \infty)$ because $f'' (2)$ is positive.

Concave up on $[0, \infty)$ since $f'' (x)$ is positive

Step 5