Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

To write $\frac{1}{{(x + h)}^{3}}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

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

Step 1.2

To write $\frac{1}{x^{3}}$ as a fraction with a common denominator, multiply by $\frac{{(x + h)}^{3}}{{(x + h)}^{3}}$ .

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

Step 1.3

Write each expression with a common denominator of ${(x + h)}^{3} x^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{{(x + h)}^{3}}$ by $\frac{x^{3}}{x^{3}}$ .

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

Step 1.3.2

Multiply $\frac{1}{x^{3}}$ by $\frac{{(x + h)}^{3}}{{(x + h)}^{3}}$ .

$\frac{\frac{x^{3}}{{(x + h)}^{3} x^{3}} + \frac{{(x + h)}^{3}}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.3.3

Reorder the factors of ${(x + h)}^{3} x^{3}$ .

$\frac{\frac{x^{3}}{x^{3} {(x + h)}^{3}} + \frac{{(x + h)}^{3}}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{x^{3} + {(x + h)}^{3}}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5

Rewrite $\frac{x^{3} + {(x + h)}^{3}}{x^{3} {(x + h)}^{3}}$ in a factored form.

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Step 1.5.1

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = x + h$ .

$\frac{\frac{(x + x + h) (x^{2} - x (x + h) + {(x + h)}^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2

Simplify.

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Step 1.5.2.1

Add $x$ and $x$ .

$\frac{\frac{(2 x + h) (x^{2} - x (x + h) + {(x + h)}^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.2

Apply the distributive property.

$\frac{\frac{(2 x + h) (x^{2} - x \cdot x - x h + {(x + h)}^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.5.2.3.1

Move $x$ .

$\frac{\frac{(2 x + h) (x^{2} - (x \cdot x) - x h + {(x + h)}^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.3.2

Multiply $x$ by $x$ .

$\frac{\frac{(2 x + h) (x^{2} - x^{2} - x h + {(x + h)}^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.4

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$\frac{\frac{(2 x + h) (x^{2} - x^{2} - x h + (x + h) (x + h))}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.5

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 1.5.2.5.1

Apply the distributive property.

$\frac{\frac{(2 x + h) (x^{2} - x^{2} - x h + x (x + h) + h (x + h))}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.5.2

Apply the distributive property.

$\frac{\frac{(2 x + h) (x^{2} - x^{2} - x h + x \cdot x + x h + h (x + h))}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.5.3

Apply the distributive property.

$\frac{\frac{(2 x + h) (x^{2} - x^{2} - x h + x \cdot x + x h + h x + h \cdot h)}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.6

Simplify and combine like terms.

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Step 1.5.2.6.1

Simplify each term.

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Step 1.5.2.6.1.1

Multiply $x$ by $x$ .

$\frac{\frac{(2 x + h) (x^{2} - x^{2} - x h + x^{2} + x h + h x + h \cdot h)}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.6.1.2

Multiply $h$ by $h$ .

$\frac{\frac{(2 x + h) (x^{2} - x^{2} - x h + x^{2} + x h + h x + h^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.6.2

Add $x h$ and $h x$ .

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Step 1.5.2.6.2.1

Reorder $x$ and $h$ .

$\frac{\frac{(2 x + h) (x^{2} - x^{2} - x h + x^{2} + h x + h x + h^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.6.2.2

Add $h x$ and $h x$ .

$\frac{\frac{(2 x + h) (x^{2} - x^{2} - x h + x^{2} + 2 h x + h^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.7

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

$\frac{\frac{(2 x + h) (0 - x h + x^{2} + 2 h x + h^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.8

Subtract $x h$ from $0$ .

$\frac{\frac{(2 x + h) (- x h + x^{2} + 2 h x + h^{2})}{x^{3} {(x + h)}^{3}}}{h}$

Step 1.5.2.9

Add $- x h$ and $2 h x$ .

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Step 1.5.2.9.1

Move $x$ .

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

Step 1.5.2.9.2

Add $- 1 h x$ and $2 h x$ .

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

Step 1.5.2.10

Multiply $h$ by $1$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Combine.

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

Step 4

Simplify the expression.

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Step 4.1

Multiply $2 x + h$ by $1$ .

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

Step 4.2

Reorder factors in $\frac{(2 x + h) (x^{2} + h x + h^{2})}{x^{3} {(x + h)}^{3} h}$ .

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