Calculus Examples

$\frac{\frac{1}{x} - \frac{1}{5}}{\frac{1}{x^{2}} - \frac{1}{25}}$

Step 1

Multiply the numerator and denominator of the fraction by $25 x^{2}$ .

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Step 1.1

Multiply $\frac{\frac{1}{x} - \frac{1}{5}}{\frac{1}{x^{2}} - \frac{1}{25}}$ by $\frac{25 x^{2}}{25 x^{2}}$ .

$\frac{25 x^{2}}{25 x^{2}} \cdot \frac{\frac{1}{x} - \frac{1}{5}}{\frac{1}{x^{2}} - \frac{1}{25}}$

Step 1.2

Combine.

$\frac{25 x^{2} (\frac{1}{x} - \frac{1}{5})}{25 x^{2} (\frac{1}{x^{2}} - \frac{1}{25})}$

Step 2

Apply the distributive property.

$\frac{25 x^{2} \frac{1}{x} + 25 x^{2} (- \frac{1}{5})}{25 x^{2} \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $25 x^{2}$ .

$\frac{x (25 x) \frac{1}{x} + 25 x^{2} (- \frac{1}{5})}{25 x^{2} \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.1.2

Cancel the common factor.

$\frac{x (25 x) \frac{1}{x} + 25 x^{2} (- \frac{1}{5})}{25 x^{2} \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.1.3

Rewrite the expression.

$\frac{25 x + 25 x^{2} (- \frac{1}{5})}{25 x^{2} \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.2

Cancel the common factor of $5$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$\frac{25 x + 25 x^{2} \frac{- 1}{5}}{25 x^{2} \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.2.2

Factor $5$ out of $25 x^{2}$ .

$\frac{25 x + 5 (5 x^{2}) \frac{- 1}{5}}{25 x^{2} \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.2.3

Cancel the common factor.

$\frac{25 x + 5 (5 x^{2}) \frac{- 1}{5}}{25 x^{2} \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.2.4

Rewrite the expression.

$\frac{25 x + 5 x^{2} \cdot - 1}{25 x^{2} \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.3

Multiply $- 1$ by $5$ .

$\frac{25 x - 5 x^{2}}{25 x^{2} \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.4

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

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Step 3.4.1

Factor $x^{2}$ out of $25 x^{2}$ .

$\frac{25 x - 5 x^{2}}{x^{2} \cdot 25 \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.4.2

Cancel the common factor.

$\frac{25 x - 5 x^{2}}{x^{2} \cdot 25 \frac{1}{x^{2}} + 25 x^{2} (- \frac{1}{25})}$

Step 3.4.3

Rewrite the expression.

$\frac{25 x - 5 x^{2}}{25 + 25 x^{2} (- \frac{1}{25})}$

Step 3.5

Cancel the common factor of $25$ .

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Step 3.5.1

Move the leading negative in $- \frac{1}{25}$ into the numerator.

$\frac{25 x - 5 x^{2}}{25 + 25 x^{2} \frac{- 1}{25}}$

Step 3.5.2

Factor $25$ out of $25 x^{2}$ .

$\frac{25 x - 5 x^{2}}{25 + 25 (x^{2}) \frac{- 1}{25}}$

Step 3.5.3

Cancel the common factor.

$\frac{25 x - 5 x^{2}}{25 + 25 x^{2} \frac{- 1}{25}}$

Step 3.5.4

Rewrite the expression.

$\frac{25 x - 5 x^{2}}{25 + x^{2} \cdot - 1}$

Step 4

Factor $5 x$ out of $25 x - 5 x^{2}$ .

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Step 4.1

Factor $5 x$ out of $25 x$ .

$\frac{5 x (5) - 5 x^{2}}{25 + x^{2} \cdot - 1}$

Step 4.2

Factor $5 x$ out of $- 5 x^{2}$ .

$\frac{5 x (5) + 5 x (- x)}{25 + x^{2} \cdot - 1}$

Step 4.3

Factor $5 x$ out of $5 x (5) + 5 x (- x)$ .

$\frac{5 x (5 - x)}{25 + x^{2} \cdot - 1}$

Step 5

Simplify the denominator.

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Step 5.1

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

$\frac{5 x (5 - x)}{25 - 1 \cdot x^{2}}$

Step 5.2

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

$\frac{5 x (5 - x)}{25 - x^{2}}$

Step 5.3

Rewrite $25 - x^{2}$ in a factored form.

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Step 5.3.1

Rewrite $25$ as $5^{2}$ .

$\frac{5 x (5 - x)}{5^{2} - x^{2}}$

Step 5.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$\frac{5 x (5 - x)}{(5 + x) (5 - x)}$

Step 6

Cancel the common factor of $5 - x$ .

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Step 6.1

Cancel the common factor.

$\frac{5 x (5 - x)}{(5 + x) (5 - x)}$

Step 6.2

Rewrite the expression.

$\frac{5 x}{5 + x}$