Calculus Examples

$f (x) = \frac{1 + \sqrt{3 x}}{1 - \sqrt{3 x}}$

Step 1

Find the first derivative.

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Step 1.1

Simplify with factoring out.

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Step 1.1.1

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

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

Step 1.1.2

Factor $3$ out of $3 x$ .

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

Step 1.1.3

Apply basic rules of exponents.

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Step 1.1.3.1

Apply the product rule to $3 (x)$ .

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

Step 1.1.3.2

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

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

Step 1.1.4

Factor $3$ out of $3 x$ .

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

Step 1.1.5

Apply the product rule to $3 (x)$ .

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

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

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

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $1 + 3^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [3^{\frac{1}{2}} x^{\frac{1}{2}}]$ .

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

Step 1.3.2

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

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

Step 1.3.3

Add $0$ and $\frac{d}{d x} [3^{\frac{1}{2}} x^{\frac{1}{2}}]$ .

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

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

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

Step 1.9

By the Sum Rule, the derivative of $1 - 3^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 3^{\frac{1}{2}} x^{\frac{1}{2}}]$ .

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

Step 1.10

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

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

Step 1.11

Add $0$ and $\frac{d}{d x} [- 3^{\frac{1}{2}} x^{\frac{1}{2}}]$ .

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

Step 1.12

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

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

Step 1.13

Multiply.

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Step 1.13.1

Multiply $- 1$ by $- 1$ .

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

Step 1.13.2

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

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

Step 1.14

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

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

Step 1.15

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

Step 1.16

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

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

Step 1.17

Combine the numerators over the common denominator.

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

Step 1.18

Simplify the numerator.

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Step 1.18.1

Multiply $- 1$ by $2$ .

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

Step 1.18.2

Subtract $2$ from $1$ .

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

Step 1.19

Move the negative in front of the fraction.

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

Step 1.20

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

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

Step 1.21

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

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

Step 1.22

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

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

Step 1.23

Simplify.

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Step 1.23.1

Apply the distributive property.

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

Step 1.23.2

Apply the distributive property.

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

Step 1.23.3

Simplify the numerator.

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Step 1.23.3.1

Combine the opposite terms in $1 \frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - 3^{\frac{1}{2}} x^{\frac{1}{2}} \frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1 \frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 3^{\frac{1}{2}} x^{\frac{1}{2}} \frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

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Step 1.23.3.1.1

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

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

Step 1.23.3.1.2

Add $1 \frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1 \frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ and $0$ .

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

Step 1.23.3.2

Simplify each term.

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Step 1.23.3.2.1

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

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

Step 1.23.3.2.2

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

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

Step 1.23.3.3

Combine the numerators over the common denominator.

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

Step 1.23.3.4

Add $3^{\frac{1}{2}}$ and $3^{\frac{1}{2}}$ .

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

Step 1.23.3.5

Cancel the common factor.

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

Step 1.23.3.6

Rewrite the expression.

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

Step 1.23.4

Combine terms.

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Step 1.23.4.1

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

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

Step 1.23.4.2

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

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

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{1}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{1}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u_{1}$ with $x^{\frac{1}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2}$ .

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

Step 2.3

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

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

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 1 - 3^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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Step 2.4.1

To apply the Chain Rule, set $u_{2}$ as $1 - 3^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 2.4.2

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

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

Step 2.4.3

Replace all occurrences of $u_{2}$ with $1 - 3^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of $1 - 3^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 3^{\frac{1}{2}} x^{\frac{1}{2}}]$ .

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

Step 2.5.2

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

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

Step 2.5.3

Add $0$ and $\frac{d}{d x} [- 3^{\frac{1}{2}} x^{\frac{1}{2}}]$ .

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

Step 2.5.4

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

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

Step 2.5.5

Multiply $- 1$ by $2$ .

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

Step 2.5.6

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

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

Step 2.9.2

Subtract $2$ from $1$ .

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

Step 2.10

Move the negative in front of the fraction.

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Cancel the common factors.

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Step 2.17.1

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

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

Step 2.17.2

Cancel the common factor.

Step 2.17.3

Rewrite the expression.

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

Step 2.18

Simplify terms.

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Step 2.18.1

Move the negative in front of the fraction.

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

Step 2.18.2

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

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

Step 2.18.3

Cancel the common factor.

Step 2.18.4

Divide $3^{\frac{1}{2}}$ by $1$ .

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

Step 2.19

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

Step 2.20

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

Step 2.21

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

Step 2.22

Combine the numerators over the common denominator.

Step 2.23

Simplify the numerator.

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Step 2.23.1

Multiply $- 1$ by $2$ .

Step 2.23.2

Subtract $2$ from $1$ .

Step 2.24

Move the negative in front of the fraction.

Step 2.25

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

Step 2.26

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

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

Step 2.27

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

Step 2.28

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

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

Step 2.29

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

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

Step 2.30

Combine the numerators over the common denominator.

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

Step 2.31

Multiply $2$ by $- 1$ .

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

Step 2.32

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

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

Step 2.33

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

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

Step 2.34

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

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

Step 2.35

Simplify.

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Step 2.35.1

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

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

Step 2.35.2

Apply the distributive property.

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

Step 2.35.3

Apply the distributive property.

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

Step 2.35.4

Apply the distributive property.

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

Step 2.35.5

Simplify the numerator.

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Step 2.35.5.1

Simplify each term.

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Step 2.35.5.1.1

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

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Step 2.35.5.1.1.1

Move $3^{\frac{1}{2}}$ .

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

Step 2.35.5.1.1.2

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

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

Step 2.35.5.1.1.3

Combine the numerators over the common denominator.

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

Step 2.35.5.1.1.4

Add $1$ and $1$ .

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

Step 2.35.5.1.1.5

Divide $2$ by $2$ .

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

Step 2.35.5.1.2

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

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

Step 2.35.5.1.3

Multiply $- 2$ by $3$ .

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

Step 2.35.5.1.4

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

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Step 2.35.5.1.4.1

Move $3^{\frac{1}{2}}$ .

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

Step 2.35.5.1.4.2

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

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

Step 2.35.5.1.4.3

Combine the numerators over the common denominator.

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

Step 2.35.5.1.4.4

Add $1$ and $1$ .

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

Step 2.35.5.1.4.5

Divide $2$ by $2$ .

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

Step 2.35.5.1.5

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

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

Step 2.35.5.1.6

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

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Step 2.35.5.1.6.1

Move $3^{\frac{1}{2}}$ .

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

Step 2.35.5.1.6.2

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

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Step 2.35.5.1.6.2.1

Raise $3$ to the power of $1$ .

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

Step 2.35.5.1.6.2.2

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

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

Step 2.35.5.1.6.3

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

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

Step 2.35.5.1.6.4

Combine the numerators over the common denominator.

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

Step 2.35.5.1.6.5

Add $1$ and $2$ .

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

Step 2.35.5.1.7

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

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Step 2.35.5.1.7.1

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

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

Step 2.35.5.1.7.2

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

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

Step 2.35.5.1.7.3

Combine the numerators over the common denominator.

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

Step 2.35.5.1.7.4

Add $1$ and $1$ .

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

Step 2.35.5.1.7.5

Divide $2$ by $2$ .

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

Step 2.35.5.1.8

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

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

Step 2.35.5.1.9

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

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

Step 2.35.5.1.10

Multiply $- 1$ by $- 2$ .

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

Step 2.35.5.1.11

Move $2$ to the left of $3^{\frac{3}{2}}$ .

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

Step 2.35.5.1.12

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

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

Step 2.35.5.1.13

Expand $(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}}) (1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})$ using the FOIL Method.

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Step 2.35.5.1.13.1

Apply the distributive property.

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

Step 2.35.5.1.13.2

Apply the distributive property.

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

Step 2.35.5.1.13.3

Apply the distributive property.

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

Step 2.35.5.1.14

Simplify and combine like terms.

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Step 2.35.5.1.14.1

Simplify each term.

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Step 2.35.5.1.14.1.1

Multiply $1$ by $1$ .

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

Step 2.35.5.1.14.1.2

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

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

Step 2.35.5.1.14.1.3

Multiply $- 1$ by $1$ .

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

Step 2.35.5.1.14.1.4

Rewrite $- 1 \cdot 3^{\frac{1}{2}}$ as $- 3^{\frac{1}{2}}$ .

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

Step 2.35.5.1.14.1.5

Rewrite using the commutative property of multiplication.

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

Step 2.35.5.1.14.1.6

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

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Step 2.35.5.1.14.1.6.1

Move $3^{\frac{1}{2}}$ .

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

Step 2.35.5.1.14.1.6.2

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

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

Step 2.35.5.1.14.1.6.3

Combine the numerators over the common denominator.

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

Step 2.35.5.1.14.1.6.4

Add $1$ and $1$ .

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

Step 2.35.5.1.14.1.6.5

Divide $2$ by $2$ .

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

Step 2.35.5.1.14.1.7

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

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

Step 2.35.5.1.14.1.8

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

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Step 2.35.5.1.14.1.8.1

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

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

Step 2.35.5.1.14.1.8.2

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

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

Step 2.35.5.1.14.1.8.3

Combine the numerators over the common denominator.

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

Step 2.35.5.1.14.1.8.4

Add $1$ and $1$ .

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

Step 2.35.5.1.14.1.8.5

Divide $2$ by $2$ .

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

Step 2.35.5.1.14.1.9

Simplify $- 1 \cdot 3 \cdot - 1 x^{1}$ .

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

Step 2.35.5.1.14.1.10

Multiply $- 1 \cdot 3 \cdot - 1$ .

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Step 2.35.5.1.14.1.10.1

Multiply $- 1$ by $3$ .

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

Step 2.35.5.1.14.1.10.2

Multiply $- 3$ by $- 1$ .

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

Step 2.35.5.1.14.2

Subtract $3^{\frac{1}{2}} x^{\frac{1}{2}}$ from $- 3^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 2.35.5.1.15

Apply the distributive property.

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

Step 2.35.5.1.16

Simplify.

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Step 2.35.5.1.16.1

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

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

Step 2.35.5.1.16.2

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

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Step 2.35.5.1.16.2.1

Move $3^{\frac{1}{2}}$ .

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

Step 2.35.5.1.16.2.2

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

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

Step 2.35.5.1.16.2.3

Combine the numerators over the common denominator.

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

Step 2.35.5.1.16.2.4

Add $1$ and $1$ .

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

Step 2.35.5.1.16.2.5

Divide $2$ by $2$ .

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

Step 2.35.5.1.16.3

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

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

Step 2.35.5.1.16.4

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

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Step 2.35.5.1.16.4.1

Move $3$ .

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

Step 2.35.5.1.16.4.2

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

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Step 2.35.5.1.16.4.2.1

Raise $3$ to the power of $1$ .

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

Step 2.35.5.1.16.4.2.2

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

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

Step 2.35.5.1.16.4.3

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

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

Step 2.35.5.1.16.4.4

Combine the numerators over the common denominator.

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

Step 2.35.5.1.16.4.5

Add $2$ and $1$ .

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

Step 2.35.5.1.17

Multiply $- 2$ by $3$ .

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

Step 2.35.5.2

Subtract $6 x^{\frac{1}{2}}$ from $- 6 x^{\frac{1}{2}}$ .

$- \frac{- 12 x^{\frac{1}{2}} + 2 \cdot 3^{\frac{3}{2}} x + 3^{\frac{1}{2}} + 3^{\frac{3}{2}} x}{2 x^{\frac{1}{2}} ({(x^{\frac{1}{2}})}^{2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.5.3

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

$- \frac{- 12 x^{\frac{1}{2}} + 3 \cdot 3^{\frac{3}{2}} x + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} ({(x^{\frac{1}{2}})}^{2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.5.4

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

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Step 2.35.5.4.1

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

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Step 2.35.5.4.1.1

Raise $3$ to the power of $1$ .

$- \frac{- 12 x^{\frac{1}{2}} + 3^{1} \cdot 3^{\frac{3}{2}} x + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} ({(x^{\frac{1}{2}})}^{2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.5.4.1.2

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

$- \frac{- 12 x^{\frac{1}{2}} + 3^{1 + \frac{3}{2}} x + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} ({(x^{\frac{1}{2}})}^{2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.5.4.2

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

$- \frac{- 12 x^{\frac{1}{2}} + 3^{\frac{2}{2} + \frac{3}{2}} x + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} ({(x^{\frac{1}{2}})}^{2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.5.4.3

Combine the numerators over the common denominator.

$- \frac{- 12 x^{\frac{1}{2}} + 3^{\frac{2 + 3}{2}} x + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} ({(x^{\frac{1}{2}})}^{2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.5.4.4

Add $2$ and $3$ .

$- \frac{- 12 x^{\frac{1}{2}} + 3^{\frac{5}{2}} x + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} ({(x^{\frac{1}{2}})}^{2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.5.5

Reorder factors in $- 12 x^{\frac{1}{2}} + 3^{\frac{5}{2}} x + 3^{\frac{1}{2}}$ .

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} ({(x^{\frac{1}{2}})}^{2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.6

Combine terms.

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Step 2.35.6.1

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

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Step 2.35.6.1.1

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

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} (x^{\frac{1}{2} \cdot 2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.6.1.2

Cancel the common factor of $2$ .

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Step 2.35.6.1.2.1

Cancel the common factor.

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} (x^{\frac{1}{2} \cdot 2} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.6.1.2.2

Rewrite the expression.

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} (x^{1} {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.6.2

Simplify.

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} (x {({(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2})}^{2})}$

Step 2.35.6.3

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

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Step 2.35.6.3.1

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

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} (x {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2 \cdot 2})}$

Step 2.35.6.3.2

Multiply $2$ by $2$ .

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} (x {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4})}$

Step 2.35.6.4

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

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 (x^{1} x^{\frac{1}{2}}) {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$

Step 2.35.6.5

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

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{1 + \frac{1}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$

Step 2.35.6.6

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

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{2}{2} + \frac{1}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$

Step 2.35.6.7

Combine the numerators over the common denominator.

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{2 + 1}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$

Step 2.35.6.8

Add $2$ and $1$ .

$- \frac{- 12 x^{\frac{1}{2}} + x \cdot 3^{\frac{5}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{3}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$

Step 2.35.7

Reorder terms.

$- \frac{3^{\frac{5}{2}} x - 12 x^{\frac{1}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{3}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$

Step 2.35.8

Reorder factors in $- \frac{3^{\frac{5}{2}} x - 12 x^{\frac{1}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{3}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$ .

$f'' (x) = - \frac{x \cdot 3^{\frac{5}{2}} - 12 x^{\frac{1}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{3}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{x \cdot 3^{\frac{5}{2}} - 12 x^{\frac{1}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{3}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$ .

$- \frac{x \cdot 3^{\frac{5}{2}} - 12 x^{\frac{1}{2}} + 3^{\frac{1}{2}}}{2 x^{\frac{3}{2}} {(1 - 3^{\frac{1}{2}} x^{\frac{1}{2}})}^{4}}$