Calculus Examples

$f (x) = - \frac{x}{7 x^{2} + 1}$

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

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 $g (x) = \frac{x}{7 x^{2} + 1}$ .

$- \frac{d}{d x} [\frac{x}{7 x^{2} + 1}] + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 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 $f (x) = x$ and $g (x) = 7 x^{2} + 1$ .

$- \frac{(7 x^{2} + 1) \frac{d}{d x} [x] - x \frac{d}{d x} [7 x^{2} + 1]}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$- \frac{(7 x^{2} + 1) \cdot 1 - x \frac{d}{d x} [7 x^{2} + 1]}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.3.2

Multiply $7 x^{2} + 1$ by $1$ .

$- \frac{7 x^{2} + 1 - x \frac{d}{d x} [7 x^{2} + 1]}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.3.3

By the Sum Rule, the derivative of $7 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [1]$ .

$- \frac{7 x^{2} + 1 - x (\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [1])}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.3.4

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$- \frac{7 x^{2} + 1 - x (7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.3.5

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

$- \frac{7 x^{2} + 1 - x (7 (2 x) + \frac{d}{d x} [1])}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.3.6

Multiply $2$ by $7$ .

$- \frac{7 x^{2} + 1 - x (14 x + \frac{d}{d x} [1])}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.3.7

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

$- \frac{7 x^{2} + 1 - x (14 x + 0)}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.3.8

Simplify the expression.

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Step 1.1.3.8.1

Add $14 x$ and $0$ .

$- \frac{7 x^{2} + 1 - x (14 x)}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.3.8.2

Multiply $14$ by $- 1$ .

$- \frac{7 x^{2} + 1 - 14 x \cdot x}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.4

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

$- \frac{7 x^{2} + 1 - 14 (x^{1} x)}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.5

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

$- \frac{7 x^{2} + 1 - 14 (x^{1} x^{1})}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.6

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

$- \frac{7 x^{2} + 1 - 14 x^{1 + 1}}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.7

Simplify by adding terms.

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Step 1.1.7.1

Add $1$ and $1$ .

$- \frac{7 x^{2} + 1 - 14 x^{2}}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.7.2

Subtract $14 x^{2}$ from $7 x^{2}$ .

$- \frac{- 7 x^{2} + 1}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \frac{d}{d x} [- 1]$

Step 1.1.8

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

$- \frac{- 7 x^{2} + 1}{{(7 x^{2} + 1)}^{2}} + \frac{x}{7 x^{2} + 1} \cdot 0$

Step 1.1.9

Simplify the expression.

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Step 1.1.9.1

Multiply $\frac{x}{7 x^{2} + 1}$ by $0$ .

$- \frac{- 7 x^{2} + 1}{{(7 x^{2} + 1)}^{2}} + 0$

Step 1.1.9.2

Add $- \frac{- 7 x^{2} + 1}{{(7 x^{2} + 1)}^{2}}$ and $0$ .

$- \frac{- 7 x^{2} + 1}{{(7 x^{2} + 1)}^{2}}$

Step 1.1.10

Simplify.

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Step 1.1.10.1

Factor $- 1$ out of $- 7 x^{2}$ .

$- \frac{- (7 x^{2}) + 1}{{(7 x^{2} + 1)}^{2}}$

Step 1.1.10.2

Rewrite $1$ as $- 1 (- 1)$ .

$- \frac{- (7 x^{2}) - 1 (- 1)}{{(7 x^{2} + 1)}^{2}}$

Step 1.1.10.3

Factor $- 1$ out of $- (7 x^{2}) - 1 (- 1)$ .

$- \frac{- (7 x^{2} - 1)}{{(7 x^{2} + 1)}^{2}}$

Step 1.1.10.4

Rewrite $- (7 x^{2} - 1)$ as $- 1 (7 x^{2} - 1)$ .

$- \frac{- 1 (7 x^{2} - 1)}{{(7 x^{2} + 1)}^{2}}$

Step 1.1.10.5

Move the negative in front of the fraction.

$- - \frac{7 x^{2} - 1}{{(7 x^{2} + 1)}^{2}}$

Step 1.1.10.6

Multiply $- 1$ by $- 1$ .

$1 \frac{7 x^{2} - 1}{{(7 x^{2} + 1)}^{2}}$

Step 1.1.10.7

Multiply $\frac{7 x^{2} - 1}{{(7 x^{2} + 1)}^{2}}$ by $1$ .

$f' (x) = \frac{7 x^{2} - 1}{{(7 x^{2} + 1)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{7 x^{2} - 1}{{(7 x^{2} + 1)}^{2}}$ .

$\frac{7 x^{2} - 1}{{(7 x^{2} + 1)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{7 x^{2} - 1}{{(7 x^{2} + 1)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{7 x^{2} - 1}{{(7 x^{2} + 1)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$7 x^{2} - 1 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Add $1$ to both sides of the equation.

$7 x^{2} = 1$

Step 2.3.2

Divide each term in $7 x^{2} = 1$ by $7$ and simplify.

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Step 2.3.2.1

Divide each term in $7 x^{2} = 1$ by $7$ .

$\frac{7 x^{2}}{7} = \frac{1}{7}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $7$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{7 x^{2}}{7} = \frac{1}{7}$

Step 2.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{7}$

Step 2.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{7}}$

Step 2.3.4

Simplify $\pm \sqrt{\frac{1}{7}}$ .

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Step 2.3.4.1

Rewrite $\sqrt{\frac{1}{7}}$ as $\frac{\sqrt{1}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{7}}$

Step 2.3.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{7}}$

Step 2.3.4.3

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 2.3.4.4

Combine and simplify the denominator.

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Step 2.3.4.4.1

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.3.4.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 2.3.4.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 2.3.4.4.4

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

$x = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 2.3.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{2}}$

Step 2.3.4.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.3.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 2.3.4.4.6.2

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

$x = \pm \frac{\sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 2.3.4.4.6.3

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

$x = \pm \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.3.4.4.6.4

Cancel the common factor of $2$ .

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Step 2.3.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.3.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{7}}{7^{1}}$

Step 2.3.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{7}}{7}$

Step 2.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{7}}{7}$

Step 2.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{7}}{7}$

Step 2.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{7}}{7}, - \frac{\sqrt{7}}{7}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $- \frac{x}{7 x^{2} + 1}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{\sqrt{7}}{7}$ .

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Step 4.1.1

Substitute $\frac{\sqrt{7}}{7}$ for $x$ .

$- \frac{\frac{\sqrt{7}}{7}}{7 {(\frac{\sqrt{7}}{7})}^{2} + 1}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 {(\frac{\sqrt{7}}{7})}^{2} + 1})$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Apply the product rule to $\frac{\sqrt{7}}{7}$ .

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{{\sqrt{7}}^{2}}{7^{2}} + 1})$

Step 4.1.2.2.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.1.2.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{{(7^{\frac{1}{2}})}^{2}}{7^{2}} + 1})$

Step 4.1.2.2.2.2

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

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7^{\frac{1}{2} \cdot 2}}{7^{2}} + 1})$

Step 4.1.2.2.2.3

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

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7^{\frac{2}{2}}}{7^{2}} + 1})$

Step 4.1.2.2.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.2.2.4.1

Cancel the common factor.

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7^{\frac{2}{2}}}{7^{2}} + 1})$

Step 4.1.2.2.2.4.2

Rewrite the expression.

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7^{1}}{7^{2}} + 1})$

Step 4.1.2.2.2.5

Evaluate the exponent.

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7}{7^{2}} + 1})$

Step 4.1.2.2.3

Raise $7$ to the power of $2$ .

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 (\frac{7}{49}) + 1})$

Step 4.1.2.2.4

Cancel the common factor of $7$ .

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Step 4.1.2.2.4.1

Factor $7$ out of $49$ .

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7}{7 (7)} + 1})$

Step 4.1.2.2.4.2

Cancel the common factor.

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7}{7 \cdot 7} + 1})$

Step 4.1.2.2.4.3

Rewrite the expression.

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{\frac{7}{7} + 1})$

Step 4.1.2.2.5

Divide $7$ by $7$ .

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{1 + 1})$

Step 4.1.2.2.6

Add $1$ and $1$ .

$- (\frac{\sqrt{7}}{7} \cdot \frac{1}{2})$

Step 4.1.2.3

Multiply $\frac{\sqrt{7}}{7} \cdot \frac{1}{2}$ .

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Step 4.1.2.3.1

Multiply $\frac{\sqrt{7}}{7}$ by $\frac{1}{2}$ .

$- \frac{\sqrt{7}}{7 \cdot 2}$

Step 4.1.2.3.2

Multiply $7$ by $2$ .

$- \frac{\sqrt{7}}{14}$

Step 4.2

Evaluate at $x = - \frac{\sqrt{7}}{7}$ .

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Step 4.2.1

Substitute $- \frac{\sqrt{7}}{7}$ for $x$ .

$- \frac{- \frac{\sqrt{7}}{7}}{7 {(- \frac{\sqrt{7}}{7})}^{2} + 1}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply the numerator by the reciprocal of the denominator.

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 {(- \frac{\sqrt{7}}{7})}^{2} + 1})$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.2.1.1

Apply the product rule to $- \frac{\sqrt{7}}{7}$ .

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 ({(- 1)}^{2} {(\frac{\sqrt{7}}{7})}^{2}) + 1})$

Step 4.2.2.2.1.2

Apply the product rule to $\frac{\sqrt{7}}{7}$ .

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 ({(- 1)}^{2} \frac{{\sqrt{7}}^{2}}{7^{2}}) + 1})$

Step 4.2.2.2.2

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

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 (1 \frac{{\sqrt{7}}^{2}}{7^{2}}) + 1})$

Step 4.2.2.2.3

Multiply $\frac{{\sqrt{7}}^{2}}{7^{2}}$ by $1$ .

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{{\sqrt{7}}^{2}}{7^{2}} + 1})$

Step 4.2.2.2.4

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.2.2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{{(7^{\frac{1}{2}})}^{2}}{7^{2}} + 1})$

Step 4.2.2.2.4.2

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

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7^{\frac{1}{2} \cdot 2}}{7^{2}} + 1})$

Step 4.2.2.2.4.3

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

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7^{\frac{2}{2}}}{7^{2}} + 1})$

Step 4.2.2.2.4.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.4.4.1

Cancel the common factor.

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7^{\frac{2}{2}}}{7^{2}} + 1})$

Step 4.2.2.2.4.4.2

Rewrite the expression.

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7^{1}}{7^{2}} + 1})$

Step 4.2.2.2.4.5

Evaluate the exponent.

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7}{7^{2}} + 1})$

Step 4.2.2.2.5

Raise $7$ to the power of $2$ .

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 (\frac{7}{49}) + 1})$

Step 4.2.2.2.6

Cancel the common factor of $7$ .

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Step 4.2.2.2.6.1

Factor $7$ out of $49$ .

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7}{7 (7)} + 1})$

Step 4.2.2.2.6.2

Cancel the common factor.

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{7 \frac{7}{7 \cdot 7} + 1})$

Step 4.2.2.2.6.3

Rewrite the expression.

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{\frac{7}{7} + 1})$

Step 4.2.2.2.7

Divide $7$ by $7$ .

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{1 + 1})$

Step 4.2.2.2.8

Add $1$ and $1$ .

$- (- \frac{\sqrt{7}}{7} \cdot \frac{1}{2})$

Step 4.2.2.3

Multiply $- \frac{\sqrt{7}}{7} \cdot \frac{1}{2}$ .

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Step 4.2.2.3.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{7}}{7}$ .

$- - \frac{\sqrt{7}}{2 \cdot 7}$

Step 4.2.2.3.2

Multiply $2$ by $7$ .

$- - \frac{\sqrt{7}}{14}$

Step 4.2.2.4

Multiply $- - \frac{\sqrt{7}}{14}$ .

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Step 4.2.2.4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{7}}{14}$

Step 4.2.2.4.2

Multiply $\frac{\sqrt{7}}{14}$ by $1$ .

$\frac{\sqrt{7}}{14}$

Step 4.3

List all of the points.

$(\frac{\sqrt{7}}{7}, - \frac{\sqrt{7}}{14}), (- \frac{\sqrt{7}}{7}, \frac{\sqrt{7}}{14})$

Step 5