Calculus Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Factor $x$ out of $x^{3}$ .

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

Step 2.1.2

Cancel the common factor.

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

Step 2.1.3

Rewrite the expression.

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

Step 2.2

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

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 1$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Divide $(x - 1) x^{2}$ by $1$ .

$(x - 1) x^{2}$

Step 4.2

Apply the distributive property.

$x \cdot x^{2} - 1 x^{2}$

Step 5

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

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Step 5.1

Multiply $x$ by $x^{2}$ .

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Step 5.1.1

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

$x^{1} x^{2} - 1 x^{2}$

Step 5.1.2

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

$x^{1 + 2} - 1 x^{2}$

Step 5.2

Add $1$ and $2$ .

$x^{3} - 1 x^{2}$

Step 6

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

$x^{3} - x^{2}$