Calculus Examples

$- \frac{1}{14} \cdot \frac{1}{\sqrt{1 - {(\frac{x}{14})}^{2}}}$

Step 1

Simplify the denominator.

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Step 1.1

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

$- \frac{1}{14} \cdot \frac{1}{\sqrt{1^{2} - {(\frac{x}{14})}^{2}}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{14}$ .

$- \frac{1}{14} \cdot \frac{1}{\sqrt{(1 + \frac{x}{14}) (1 - \frac{x}{14})}}$

Step 1.3

Simplify.

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Step 1.3.1

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

$- \frac{1}{14} \cdot \frac{1}{\sqrt{(\frac{14}{14} + \frac{x}{14}) (1 - \frac{x}{14})}}$

Step 1.3.2

Combine the numerators over the common denominator.

$- \frac{1}{14} \cdot \frac{1}{\sqrt{\frac{14 + x}{14} (1 - \frac{x}{14})}}$

Step 1.3.3

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

$- \frac{1}{14} \cdot \frac{1}{\sqrt{\frac{14 + x}{14} (\frac{14}{14} - \frac{x}{14})}}$

Step 1.3.4

Combine the numerators over the common denominator.

$- \frac{1}{14} \cdot \frac{1}{\sqrt{\frac{14 + x}{14} \cdot \frac{14 - x}{14}}}$

Step 1.4

Multiply $\frac{14 + x}{14}$ by $\frac{14 - x}{14}$ .

$- \frac{1}{14} \cdot \frac{1}{\sqrt{\frac{(14 + x) (14 - x)}{14 \cdot 14}}}$

Step 1.5

Multiply $14$ by $14$ .

$- \frac{1}{14} \cdot \frac{1}{\sqrt{\frac{(14 + x) (14 - x)}{196}}}$

Step 1.6

Rewrite $\frac{(14 + x) (14 - x)}{196}$ as ${(\frac{1}{14})}^{2} ((14 + x) (14 - x))$ .

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Step 1.6.1

Factor the perfect power $1^{2}$ out of $(14 + x) (14 - x)$ .

$- \frac{1}{14} \cdot \frac{1}{\sqrt{\frac{1^{2} ((14 + x) (14 - x))}{196}}}$

Step 1.6.2

Factor the perfect power $14^{2}$ out of $196$ .

$- \frac{1}{14} \cdot \frac{1}{\sqrt{\frac{1^{2} ((14 + x) (14 - x))}{14^{2} \cdot 1}}}$

Step 1.6.3

Rearrange the fraction $\frac{1^{2} ((14 + x) (14 - x))}{14^{2} \cdot 1}$ .

$- \frac{1}{14} \cdot \frac{1}{\sqrt{{(\frac{1}{14})}^{2} ((14 + x) (14 - x))}}$

Step 1.7

Pull terms out from under the radical.

$- \frac{1}{14} \cdot \frac{1}{\frac{1}{14} \sqrt{(14 + x) (14 - x)}}$

Step 1.8

Combine $\frac{1}{14}$ and $\sqrt{(14 + x) (14 - x)}$ .

$- \frac{1}{14} \cdot \frac{1}{\frac{\sqrt{(14 + x) (14 - x)}}{14}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{14} \cdot (1 \frac{14}{\sqrt{(14 + x) (14 - x)}})$

Step 3

Multiply $\frac{14}{\sqrt{(14 + x) (14 - x)}}$ by $1$ .

$- \frac{1}{14} \cdot \frac{14}{\sqrt{(14 + x) (14 - x)}}$

Step 4

Cancel the common factor of $14$ .

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Step 4.1

Move the leading negative in $- \frac{1}{14}$ into the numerator.

$\frac{- 1}{14} \cdot \frac{14}{\sqrt{(14 + x) (14 - x)}}$

Step 4.2

Cancel the common factor.

$\frac{- 1}{14} \cdot \frac{14}{\sqrt{(14 + x) (14 - x)}}$

Step 4.3

Rewrite the expression.

$- 1 \cdot \frac{1}{\sqrt{(14 + x) (14 - x)}}$

Step 5

Multiply $\frac{1}{\sqrt{(14 + x) (14 - x)}}$ by $\frac{\sqrt{(14 + x) (14 - x)}}{\sqrt{(14 + x) (14 - x)}}$ .

$- 1 \cdot (\frac{1}{\sqrt{(14 + x) (14 - x)}} \cdot \frac{\sqrt{(14 + x) (14 - x)}}{\sqrt{(14 + x) (14 - x)}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{1}{\sqrt{(14 + x) (14 - x)}}$ by $\frac{\sqrt{(14 + x) (14 - x)}}{\sqrt{(14 + x) (14 - x)}}$ .

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{\sqrt{(14 + x) (14 - x)} \sqrt{(14 + x) (14 - x)}}$

Step 6.2

Raise $\sqrt{(14 + x) (14 - x)}$ to the power of $1$ .

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{{\sqrt{(14 + x) (14 - x)}}^{1} \sqrt{(14 + x) (14 - x)}}$

Step 6.3

Raise $\sqrt{(14 + x) (14 - x)}$ to the power of $1$ .

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{{\sqrt{(14 + x) (14 - x)}}^{1} {\sqrt{(14 + x) (14 - x)}}^{1}}$

Step 6.4

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

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{{\sqrt{(14 + x) (14 - x)}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{{\sqrt{(14 + x) (14 - x)}}^{2}}$

Step 6.6

Rewrite ${\sqrt{(14 + x) (14 - x)}}^{2}$ as $(14 + x) (14 - x)$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(14 + x) (14 - x)}$ as ${((14 + x) (14 - x))}^{\frac{1}{2}}$ .

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{{({((14 + x) (14 - x))}^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{{((14 + x) (14 - x))}^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{{((14 + x) (14 - x))}^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{{((14 + x) (14 - x))}^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{{((14 + x) (14 - x))}^{1}}$

Step 6.6.5

Simplify.

$- 1 \cdot \frac{\sqrt{(14 + x) (14 - x)}}{(14 + x) (14 - x)}$

Step 7

Rewrite $- 1 \frac{\sqrt{(14 + x) (14 - x)}}{(14 + x) (14 - x)}$ as $- \frac{\sqrt{(14 + x) (14 - x)}}{(14 + x) (14 - x)}$ .

$- \frac{\sqrt{(14 + x) (14 - x)}}{(14 + x) (14 - x)}$