Calculus Examples

$f (x) = x \sqrt{6 x + 1}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x + 1}$ as ${(6 x + 1)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x {(6 x + 1)}^{\frac{1}{2}}]$

Step 1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(6 x + 1)}^{\frac{1}{2}}$ .

$x \frac{d}{d x} [{(6 x + 1)}^{\frac{1}{2}}] + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 6 x + 1$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $6 x + 1$ .

$x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.3.2

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

$x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.3.3

Replace all occurrences of $u$ with $6 x + 1$ .

$x (\frac{1}{2} {(6 x + 1)}^{\frac{1}{2} - 1} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.4

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

$x (\frac{1}{2} {(6 x + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.5

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

$x (\frac{1}{2} {(6 x + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.6

Combine the numerators over the common denominator.

$x (\frac{1}{2} {(6 x + 1)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

$x (\frac{1}{2} {(6 x + 1)}^{\frac{1 - 2}{2}} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.7.2

Subtract $2$ from $1$ .

$x (\frac{1}{2} {(6 x + 1)}^{\frac{- 1}{2}} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8

Combine fractions.

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Step 1.1.8.1

Move the negative in front of the fraction.

$x (\frac{1}{2} {(6 x + 1)}^{- \frac{1}{2}} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8.2

Combine $\frac{1}{2}$ and ${(6 x + 1)}^{- \frac{1}{2}}$ .

$x (\frac{{(6 x + 1)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8.3

Move ${(6 x + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{2 {(6 x + 1)}^{\frac{1}{2}}} \frac{d}{d x} [6 x + 1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8.4

Combine $\frac{1}{2 {(6 x + 1)}^{\frac{1}{2}}}$ and $x$ .

$\frac{x}{2 {(6 x + 1)}^{\frac{1}{2}}} \frac{d}{d x} [6 x + 1] + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.9

By the Sum Rule, the derivative of $6 x + 1$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [1]$ .

$\frac{x}{2 {(6 x + 1)}^{\frac{1}{2}}} (\frac{d}{d x} [6 x] + \frac{d}{d x} [1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.10

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{x}{2 {(6 x + 1)}^{\frac{1}{2}}} (6 \frac{d}{d x} [x] + \frac{d}{d x} [1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.11

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

$\frac{x}{2 {(6 x + 1)}^{\frac{1}{2}}} (6 \cdot 1 + \frac{d}{d x} [1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.12

Multiply $6$ by $1$ .

$\frac{x}{2 {(6 x + 1)}^{\frac{1}{2}}} (6 + \frac{d}{d x} [1]) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.13

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

$\frac{x}{2 {(6 x + 1)}^{\frac{1}{2}}} (6 + 0) + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14

Simplify terms.

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Step 1.1.14.1

Add $6$ and $0$ .

$\frac{x}{2 {(6 x + 1)}^{\frac{1}{2}}} \cdot 6 + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14.2

Combine $\frac{x}{2 {(6 x + 1)}^{\frac{1}{2}}}$ and $6$ .

$\frac{x \cdot 6}{2 {(6 x + 1)}^{\frac{1}{2}}} + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14.3

Move $6$ to the left of $x$ .

$\frac{6 \cdot x}{2 {(6 x + 1)}^{\frac{1}{2}}} + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14.4

Factor $2$ out of $6 x$ .

$\frac{2 (3 x)}{2 {(6 x + 1)}^{\frac{1}{2}}} + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.15

Cancel the common factors.

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Step 1.1.15.1

Factor $2$ out of $2 {(6 x + 1)}^{\frac{1}{2}}$ .

$\frac{2 (3 x)}{2 ({(6 x + 1)}^{\frac{1}{2}})} + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.15.2

Cancel the common factor.

$\frac{2 (3 x)}{2 {(6 x + 1)}^{\frac{1}{2}}} + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.15.3

Rewrite the expression.

$\frac{3 x}{{(6 x + 1)}^{\frac{1}{2}}} + {(6 x + 1)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.16

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

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

Step 1.1.17

Multiply ${(6 x + 1)}^{\frac{1}{2}}$ by $1$ .

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

Step 1.1.18

To write ${(6 x + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(6 x + 1)}^{\frac{1}{2}}}{{(6 x + 1)}^{\frac{1}{2}}}$ .

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

Step 1.1.19

Combine the numerators over the common denominator.

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

Step 1.1.20

Multiply ${(6 x + 1)}^{\frac{1}{2}}$ by ${(6 x + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.1.20.1

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

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

Step 1.1.20.2

Combine the numerators over the common denominator.

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

Step 1.1.20.3

Add $1$ and $1$ .

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

Step 1.1.20.4

Divide $2$ by $2$ .

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

Step 1.1.21

Simplify ${(6 x + 1)}^{1}$ .

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

Step 1.1.22

Add $3 x$ and $6 x$ .

$f' (x) = \frac{9 x + 1}{{(6 x + 1)}^{\frac{1}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{9 x + 1}{{(6 x + 1)}^{\frac{1}{2}}}$ .

$\frac{9 x + 1}{{(6 x + 1)}^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{9 x + 1}{{(6 x + 1)}^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{9 x + 1}{{(6 x + 1)}^{\frac{1}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$9 x + 1 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Subtract $1$ from both sides of the equation.

$9 x = - 1$

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $9$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

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

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Move the negative in front of the fraction.

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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

$\frac{9 x + 1}{\sqrt{{(6 x + 1)}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$\frac{9 x + 1}{\sqrt{6 x + 1}}$

Step 3.2

Set the denominator in $\frac{9 x + 1}{\sqrt{6 x + 1}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{6 x + 1} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

${\sqrt{6 x + 1}}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x + 1}$ as ${(6 x + 1)}^{\frac{1}{2}}$ .

${({(6 x + 1)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${({(6 x + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.1

Multiply the exponents in ${({(6 x + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.1.1

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

${(6 x + 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.1.2.1

Cancel the common factor.

${(6 x + 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.1.2.2

Rewrite the expression.

${(6 x + 1)}^{1} = 0^{2}$

Step 3.3.2.2.1.2

Simplify.

$6 x + 1 = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$6 x + 1 = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Subtract $1$ from both sides of the equation.

$6 x = - 1$

Step 3.3.3.2

Divide each term in $6 x = - 1$ by $6$ and simplify.

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Step 3.3.3.2.1

Divide each term in $6 x = - 1$ by $6$ .

$\frac{6 x}{6} = \frac{- 1}{6}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $6$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 1}{6}$

Step 3.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{6}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{6}$

Step 3.4

Set the radicand in $\sqrt{6 x + 1}$ less than $0$ to find where the expression is undefined.

$6 x + 1 < 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

Subtract $1$ from both sides of the inequality.

$6 x < - 1$

Step 3.5.2

Divide each term in $6 x < - 1$ by $6$ and simplify.

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Step 3.5.2.1

Divide each term in $6 x < - 1$ by $6$ .

$\frac{6 x}{6} < \frac{- 1}{6}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $6$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} < \frac{- 1}{6}$

Step 3.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 1}{6}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Move the negative in front of the fraction.

$x < - \frac{1}{6}$

Step 3.6

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

$x \leq - \frac{1}{6}$

$(- \infty, - \frac{1}{6}]$

$x \leq - \frac{1}{6}$

$(- \infty, - \frac{1}{6}]$

Step 4

Evaluate $x \sqrt{6 x + 1}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - \frac{1}{9}$ .

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Step 4.1.1

Substitute $- \frac{1}{9}$ for $x$ .

$(- \frac{1}{9}) \sqrt{6 (- \frac{1}{9}) + 1}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $3$ .

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Step 4.1.2.1.1

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$- \frac{1}{9} \sqrt{6 (\frac{- 1}{9}) + 1}$

Step 4.1.2.1.2

Factor $3$ out of $6$ .

$- \frac{1}{9} \sqrt{3 (2) \frac{- 1}{9} + 1}$

Step 4.1.2.1.3

Factor $3$ out of $9$ .

$- \frac{1}{9} \sqrt{3 \cdot 2 \frac{- 1}{3 \cdot 3} + 1}$

Step 4.1.2.1.4

Cancel the common factor.

$- \frac{1}{9} \sqrt{3 \cdot 2 \frac{- 1}{3 \cdot 3} + 1}$

Step 4.1.2.1.5

Rewrite the expression.

$- \frac{1}{9} \sqrt{2 (\frac{- 1}{3}) + 1}$

Step 4.1.2.2

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

$- \frac{1}{9} \sqrt{\frac{2 \cdot - 1}{3} + 1}$

Step 4.1.2.3

Simplify the expression.

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Step 4.1.2.3.1

Multiply $2$ by $- 1$ .

$- \frac{1}{9} \sqrt{\frac{- 2}{3} + 1}$

Step 4.1.2.3.2

Move the negative in front of the fraction.

$- \frac{1}{9} \sqrt{- \frac{2}{3} + 1}$

Step 4.1.2.3.3

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

$- \frac{1}{9} \sqrt{- \frac{2}{3} + \frac{3}{3}}$

Step 4.1.2.3.4

Combine the numerators over the common denominator.

$- \frac{1}{9} \sqrt{\frac{- 2 + 3}{3}}$

Step 4.1.2.3.5

Add $- 2$ and $3$ .

$- \frac{1}{9} \sqrt{\frac{1}{3}}$

Step 4.1.2.4

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$- \frac{1}{9} \cdot \frac{\sqrt{1}}{\sqrt{3}}$

Step 4.1.2.5

Any root of $1$ is $1$ .

$- \frac{1}{9} \cdot \frac{1}{\sqrt{3}}$

Step 4.1.2.6

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- \frac{1}{9} (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 4.1.2.7

Combine and simplify the denominator.

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Step 4.1.2.7.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.1.2.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.1.2.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.1.2.7.4

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

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.1.2.7.5

Add $1$ and $1$ .

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.1.2.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.1.2.7.6.1

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

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.1.2.7.6.2

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

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.1.2.7.6.3

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

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.1.2.7.6.4

Cancel the common factor of $2$ .

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Step 4.1.2.7.6.4.1

Cancel the common factor.

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.1.2.7.6.4.2

Rewrite the expression.

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{3^{1}}$

Step 4.1.2.7.6.5

Evaluate the exponent.

$- \frac{1}{9} \cdot \frac{\sqrt{3}}{3}$

Step 4.1.2.8

Multiply $- \frac{1}{9} \cdot \frac{\sqrt{3}}{3}$ .

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Step 4.1.2.8.1

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{1}{9}$ .

$- \frac{\sqrt{3}}{3 \cdot 9}$

Step 4.1.2.8.2

Multiply $3$ by $9$ .

$- \frac{\sqrt{3}}{27}$

Step 4.2

Evaluate at $x = - \frac{1}{6}$ .

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Step 4.2.1

Substitute $- \frac{1}{6}$ for $x$ .

$(- \frac{1}{6}) \sqrt{6 (- \frac{1}{6}) + 1}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $6$ .

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Step 4.2.2.1.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$- \frac{1}{6} \sqrt{6 (\frac{- 1}{6}) + 1}$

Step 4.2.2.1.2

Cancel the common factor.

$- \frac{1}{6} \sqrt{6 (\frac{- 1}{6}) + 1}$

Step 4.2.2.1.3

Rewrite the expression.

$- \frac{1}{6} \sqrt{- 1 + 1}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Add $- 1$ and $1$ .

$- \frac{1}{6} \sqrt{0}$

Step 4.2.2.2.2

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

$- \frac{1}{6} \sqrt{0^{2}}$

Step 4.2.2.2.3

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{1}{6} \cdot 0$

Step 4.2.2.3

Multiply $- \frac{1}{6} \cdot 0$ .

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Step 4.2.2.3.1

Multiply $0$ by $- 1$ .

$0 (\frac{1}{6})$

Step 4.2.2.3.2

Multiply $0$ by $\frac{1}{6}$ .

$0$

Step 4.3

List all of the points.

$(- \frac{1}{9}, - \frac{\sqrt{3}}{27}), (- \frac{1}{6}, 0)$

Step 5