Finite Math Examples

$f (x) = \frac{x^{3} - 729}{x^{2} - 81}$ , $x = 9$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \frac{{(x + h)}^{3} - 729}{{(x + h)}^{2} - 81}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the numerator.

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Step 2.1.2.1.1

Rewrite $729$ as $9^{3}$ .

$f (x + h) = \frac{{(x + h)}^{3} - 9^{3}}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x + h$ and $b = 9$ .

$f (x + h) = \frac{(x + h - 9) ({(x + h)}^{2} + (x + h) \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3

Simplify.

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Step 2.1.2.1.3.1

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

$f (x + h) = \frac{(x + h - 9) ((x + h) (x + h) + (x + h) \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.2

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

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Step 2.1.2.1.3.2.1

Apply the distributive property.

$f (x + h) = \frac{(x + h - 9) (x (x + h) + h (x + h) + (x + h) \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.2.2

Apply the distributive property.

$f (x + h) = \frac{(x + h - 9) (x \cdot x + x h + h (x + h) + (x + h) \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.2.3

Apply the distributive property.

$f (x + h) = \frac{(x + h - 9) (x \cdot x + x h + h x + h \cdot h + (x + h) \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.3

Simplify and combine like terms.

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Step 2.1.2.1.3.3.1

Simplify each term.

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Step 2.1.2.1.3.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + x h + h x + h \cdot h + (x + h) \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + x h + h x + h^{2} + (x + h) \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.3.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + h x + h x + h^{2} + (x + h) \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + 2 h x + h^{2} + (x + h) \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.4

Apply the distributive property.

$f (x + h) = \frac{(x + h - 9) (x^{2} + 2 h x + h^{2} + x \cdot 9 + h \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.5

Move $9$ to the left of $x$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + 2 h x + h^{2} + 9 \cdot x + h \cdot 9 + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.6

Move $9$ to the left of $h$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + 2 h x + h^{2} + 9 \cdot x + 9 \cdot h + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.7

Multiply $9$ by $h$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + 2 h x + h^{2} + 9 x + 9 h + 9^{2})}{{(x + h)}^{2} - 81}$

Step 2.1.2.1.3.8

Raise $9$ to the power of $2$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81)}{{(x + h)}^{2} - 81}$

Step 2.1.2.2

Simplify the denominator.

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Step 2.1.2.2.1

Rewrite $81$ as $9^{2}$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81)}{{(x + h)}^{2} - 9^{2}}$

Step 2.1.2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + h$ and $b = 9$ .

$f (x + h) = \frac{(x + h - 9) (x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81)}{(x + h + 9) (x + h - 9)}$

Step 2.1.2.3

Cancel the common factor of $x + h - 9$ .

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Step 2.1.2.3.1

Cancel the common factor.

$f (x + h) = \frac{(x + h - 9) (x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81)}{(x + h + 9) (x + h - 9)}$

Step 2.1.2.3.2

Rewrite the expression.

$f (x + h) = \frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9}$

Step 2.1.2.4

The final answer is $\frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9}$ .

$\frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9}$

$f (x) = \frac{x^{2} + 9 x + 81}{x + 9}$

$f (x + h) = \frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9}$

$f (x) = \frac{x^{2} + 9 x + 81}{x + 9}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9} - (\frac{x^{2} + 9 x + 81}{x + 9})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9}$ as a fraction with a common denominator, multiply by $\frac{x + 9}{x + 9}$ .

$\frac{\frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9} \cdot \frac{x + 9}{x + 9} - \frac{x^{2} + 9 x + 81}{x + 9}}{h}$

Step 4.1.2

To write $- \frac{x^{2} + 9 x + 81}{x + 9}$ as a fraction with a common denominator, multiply by $\frac{x + h + 9}{x + h + 9}$ .

$\frac{\frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9} \cdot \frac{x + 9}{x + 9} - \frac{x^{2} + 9 x + 81}{x + 9} \cdot \frac{x + h + 9}{x + h + 9}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(x + h + 9) (x + 9)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81}{x + h + 9}$ by $\frac{x + 9}{x + 9}$ .

$\frac{\frac{(x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81) (x + 9)}{(x + h + 9) (x + 9)} - \frac{x^{2} + 9 x + 81}{x + 9} \cdot \frac{x + h + 9}{x + h + 9}}{h}$

Step 4.1.3.2

Multiply $\frac{x^{2} + 9 x + 81}{x + 9}$ by $\frac{x + h + 9}{x + h + 9}$ .

$\frac{\frac{(x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81) (x + 9)}{(x + h + 9) (x + 9)} - \frac{(x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.3.3

Reorder the factors of $(x + h + 9) (x + 9)$ .

$\frac{\frac{(x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81) (x + 9)}{(x + 9) (x + h + 9)} - \frac{(x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{(x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81) (x + 9) - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5

Rewrite $\frac{(x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81) (x + 9) - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}$ in a factored form.

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Step 4.1.5.1

Expand $(x^{2} + 2 h x + h^{2} + 9 x + 9 h + 81) (x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{x^{2} x + x^{2} \cdot 9 + 2 h x \cdot x + 2 h x \cdot 9 + h^{2} x + h^{2} \cdot 9 + 9 x \cdot x + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2

Simplify each term.

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Step 4.1.5.2.1

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

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Step 4.1.5.2.1.1

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

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Step 4.1.5.2.1.1.1

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

$\frac{\frac{x^{2} x^{1} + x^{2} \cdot 9 + 2 h x \cdot x + 2 h x \cdot 9 + h^{2} x + h^{2} \cdot 9 + 9 x \cdot x + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.1.1.2

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

$\frac{\frac{x^{2 + 1} + x^{2} \cdot 9 + 2 h x \cdot x + 2 h x \cdot 9 + h^{2} x + h^{2} \cdot 9 + 9 x \cdot x + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.1.2

Add $2$ and $1$ .

$\frac{\frac{x^{3} + x^{2} \cdot 9 + 2 h x \cdot x + 2 h x \cdot 9 + h^{2} x + h^{2} \cdot 9 + 9 x \cdot x + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.2

Move $9$ to the left of $x^{2}$ .

$\frac{\frac{x^{3} + 9 \cdot x^{2} + 2 h x \cdot x + 2 h x \cdot 9 + h^{2} x + h^{2} \cdot 9 + 9 x \cdot x + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.3

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

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Step 4.1.5.2.3.1

Move $x$ .

$\frac{\frac{x^{3} + 9 x^{2} + 2 h (x \cdot x) + 2 h x \cdot 9 + h^{2} x + h^{2} \cdot 9 + 9 x \cdot x + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.3.2

Multiply $x$ by $x$ .

$\frac{\frac{x^{3} + 9 x^{2} + 2 h x^{2} + 2 h x \cdot 9 + h^{2} x + h^{2} \cdot 9 + 9 x \cdot x + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.4

Multiply $9$ by $2$ .

$\frac{\frac{x^{3} + 9 x^{2} + 2 h x^{2} + 18 h x + h^{2} x + h^{2} \cdot 9 + 9 x \cdot x + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.5

Move $9$ to the left of $h^{2}$ .

$\frac{\frac{x^{3} + 9 x^{2} + 2 h x^{2} + 18 h x + h^{2} x + 9 \cdot h^{2} + 9 x \cdot x + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.6

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

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Step 4.1.5.2.6.1

Move $x$ .

$\frac{\frac{x^{3} + 9 x^{2} + 2 h x^{2} + 18 h x + h^{2} x + 9 h^{2} + 9 (x \cdot x) + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.6.2

Multiply $x$ by $x$ .

$\frac{\frac{x^{3} + 9 x^{2} + 2 h x^{2} + 18 h x + h^{2} x + 9 h^{2} + 9 x^{2} + 9 x \cdot 9 + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.7

Multiply $9$ by $9$ .

$\frac{\frac{x^{3} + 9 x^{2} + 2 h x^{2} + 18 h x + h^{2} x + 9 h^{2} + 9 x^{2} + 81 x + 9 h x + 9 h \cdot 9 + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.8

Multiply $9$ by $9$ .

$\frac{\frac{x^{3} + 9 x^{2} + 2 h x^{2} + 18 h x + h^{2} x + 9 h^{2} + 9 x^{2} + 81 x + 9 h x + 81 h + 81 x + 81 \cdot 9 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.2.9

Multiply $81$ by $9$ .

$\frac{\frac{x^{3} + 9 x^{2} + 2 h x^{2} + 18 h x + h^{2} x + 9 h^{2} + 9 x^{2} + 81 x + 9 h x + 81 h + 81 x + 729 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.3

Add $9 x^{2}$ and $9 x^{2}$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 18 h x + h^{2} x + 9 h^{2} + 81 x + 9 h x + 81 h + 81 x + 729 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.4

Add $18 h x$ and $9 h x$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 81 x + 81 h + 81 x + 729 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.5

Add $81 x$ and $81 x$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - (x^{2} + 9 x + 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.6

Apply the distributive property.

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 + (- x^{2} - (9 x) - 1 \cdot 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.7

Simplify.

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Step 4.1.5.7.1

Multiply $9$ by $- 1$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 + (- x^{2} - 9 x - 1 \cdot 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.7.2

Multiply $- 1$ by $81$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 + (- x^{2} - 9 x - 81) (x + h + 9)}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.8

Expand $(- x^{2} - 9 x - 81) (x + h + 9)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{2} x - x^{2} h - x^{2} \cdot 9 - 9 x \cdot x - 9 x h - 9 x \cdot 9 - 81 x - 81 h - 81 \cdot 9}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.9

Simplify each term.

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Step 4.1.5.9.1

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

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Step 4.1.5.9.1.1

Move $x$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - (x \cdot x^{2}) - x^{2} h - x^{2} \cdot 9 - 9 x \cdot x - 9 x h - 9 x \cdot 9 - 81 x - 81 h - 81 \cdot 9}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.9.1.2

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

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Step 4.1.5.9.1.2.1

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

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - (x^{1} x^{2}) - x^{2} h - x^{2} \cdot 9 - 9 x \cdot x - 9 x h - 9 x \cdot 9 - 81 x - 81 h - 81 \cdot 9}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.9.1.2.2

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

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{1 + 2} - x^{2} h - x^{2} \cdot 9 - 9 x \cdot x - 9 x h - 9 x \cdot 9 - 81 x - 81 h - 81 \cdot 9}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.9.1.3

Add $1$ and $2$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{3} - x^{2} h - x^{2} \cdot 9 - 9 x \cdot x - 9 x h - 9 x \cdot 9 - 81 x - 81 h - 81 \cdot 9}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.9.2

Multiply $9$ by $- 1$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{3} - x^{2} h - 9 x^{2} - 9 x \cdot x - 9 x h - 9 x \cdot 9 - 81 x - 81 h - 81 \cdot 9}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.9.3

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

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Step 4.1.5.9.3.1

Move $x$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{3} - x^{2} h - 9 x^{2} - 9 (x \cdot x) - 9 x h - 9 x \cdot 9 - 81 x - 81 h - 81 \cdot 9}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.9.3.2

Multiply $x$ by $x$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{3} - x^{2} h - 9 x^{2} - 9 x^{2} - 9 x h - 9 x \cdot 9 - 81 x - 81 h - 81 \cdot 9}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.9.4

Multiply $9$ by $- 9$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{3} - x^{2} h - 9 x^{2} - 9 x^{2} - 9 x h - 81 x - 81 x - 81 h - 81 \cdot 9}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.9.5

Multiply $- 81$ by $9$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{3} - x^{2} h - 9 x^{2} - 9 x^{2} - 9 x h - 81 x - 81 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.10

Subtract $9 x^{2}$ from $- 9 x^{2}$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{3} - x^{2} h - 18 x^{2} - 9 x h - 81 x - 81 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.11

Subtract $81 x$ from $- 81 x$ .

$\frac{\frac{x^{3} + 18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{3} - x^{2} h - 18 x^{2} - 9 x h - 162 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.12

Subtract $x^{3}$ from $x^{3}$ .

$\frac{\frac{18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 + 0 - x^{2} h - 18 x^{2} - 9 x h - 162 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.13

Add $18 x^{2}$ and $0$ .

$\frac{\frac{18 x^{2} + 2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{2} h - 18 x^{2} - 9 x h - 162 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.14

Subtract $18 x^{2}$ from $18 x^{2}$ .

$\frac{\frac{2 h x^{2} + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - x^{2} h + 0 - 9 x h - 162 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.15

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

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Step 4.1.5.15.1

Move $h$ .

$\frac{\frac{2 x^{2} h - x^{2} h + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 + 0 - 9 x h - 162 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.15.2

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

$\frac{\frac{1 x^{2} h + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 + 0 - 9 x h - 162 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.16

Add $1 x^{2} h$ and $0$ .

$\frac{\frac{1 x^{2} h + 27 h x + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - 9 x h - 162 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.17

Subtract $9 x h$ from $27 h x$ .

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Step 4.1.5.17.1

Move $h$ .

$\frac{\frac{1 x^{2} h + 27 x h - 9 x h + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - 162 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.17.2

Subtract $9 x h$ from $27 x h$ .

$\frac{\frac{1 x^{2} h + 18 x h + h^{2} x + 9 h^{2} + 162 x + 81 h + 729 - 162 x - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.18

Subtract $162 x$ from $162 x$ .

$\frac{\frac{1 x^{2} h + 18 x h + h^{2} x + 9 h^{2} + 81 h + 729 + 0 - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.19

Add $1 x^{2} h$ and $0$ .

$\frac{\frac{1 x^{2} h + 18 x h + h^{2} x + 9 h^{2} + 81 h + 729 - 81 h - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.20

Subtract $81 h$ from $81 h$ .

$\frac{\frac{1 x^{2} h + 18 x h + h^{2} x + 9 h^{2} + 0 + 729 - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.21

Add $1 x^{2} h$ and $0$ .

$\frac{\frac{18 x h + h^{2} x + 9 h^{2} + 1 x^{2} h + 729 - 729}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.22

Subtract $729$ from $729$ .

$\frac{\frac{18 x h + h^{2} x + 9 h^{2} + 1 x^{2} h + 0}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.23

Add $18 x h + h^{2} x + 9 h^{2} + 1 x^{2} h$ and $0$ .

$\frac{\frac{18 x h + h^{2} x + 9 h^{2} + 1 x^{2} h}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.24

Rewrite $18 x h + h^{2} x + 9 h^{2} + 1 x^{2} h$ in a factored form.

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Step 4.1.5.24.1

Factor $h$ out of $18 x h + h^{2} x + 9 h^{2} + 1 x^{2} h$ .

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Step 4.1.5.24.1.1

Factor $h$ out of $18 x h$ .

$\frac{\frac{h (18 x) + h^{2} x + 9 h^{2} + 1 x^{2} h}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.24.1.2

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

$\frac{\frac{h (18 x) + h (h (x)) + 9 h^{2} + 1 x^{2} h}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.24.1.3

Factor $h$ out of $9 h^{2}$ .

$\frac{\frac{h (18 x) + h (h (x)) + h (9 h) + 1 x^{2} h}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.24.1.4

Factor $h$ out of $1 x^{2} h$ .

$\frac{\frac{h (18 x) + h (h (x)) + h (9 h) + h (1 x^{2})}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.24.1.5

Factor $h$ out of $h (18 x) + h (h (x))$ .

$\frac{\frac{h (18 x + h (x)) + h (9 h) + h (1 x^{2})}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.24.1.6

Factor $h$ out of $h (18 x + h (x)) + h (9 h)$ .

$\frac{\frac{h (18 x + h (x) + 9 h) + h (1 x^{2})}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.24.1.7

Factor $h$ out of $h (18 x + h (x) + 9 h) + h (1 x^{2})$ .

$\frac{\frac{h (18 x + h (x) + 9 h + 1 x^{2})}{(x + 9) (x + h + 9)}}{h}$

Step 4.1.5.24.2

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

$\frac{\frac{h (18 x + h (x) + 9 h + x^{2})}{(x + 9) (x + h + 9)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{h (18 x + h (x) + 9 h + x^{2})}{(x + 9) (x + h + 9)} \cdot \frac{1}{h}$

Step 4.3

Combine.

$\frac{h (18 x + h (x) + 9 h + x^{2}) \cdot 1}{(x + 9) (x + h + 9) h}$

Step 4.4

Cancel the common factor of $h$ .

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Step 4.4.1

Cancel the common factor.

$\frac{h (18 x + h (x) + 9 h + x^{2}) \cdot 1}{(x + 9) (x + h + 9) h}$

Step 4.4.2

Rewrite the expression.

$\frac{(18 x + h x + 9 h + x^{2}) \cdot 1}{(x + 9) (x + h + 9)}$

Step 4.5

Multiply $18 x + h x + 9 h + x^{2}$ by $1$ .

$\frac{18 x + h x + 9 h + x^{2}}{(x + 9) (x + h + 9)}$

Step 5

Replace the variable $x$ with $9$ in the expression.

$\frac{18 (9) + h (9) + 9 h + {(9)}^{2}}{((9) + 9) ((9) + h + 9)}$

Step 6

Cancel the common factor of $18 (9) + h (9) + 9 h + {(9)}^{2}$ and $(9) + 9$ .

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Step 6.1

Factor $9$ out of $18 (9)$ .

$\frac{9 (2 \cdot 9) + h (9) + 9 h + {(9)}^{2}}{((9) + 9) ((9) + h + 9)}$

Step 6.2

Factor $9$ out of $h (9)$ .

$\frac{9 (2 \cdot 9) + 9 h + 9 h + {(9)}^{2}}{((9) + 9) ((9) + h + 9)}$

Step 6.3

Factor $9$ out of $9 (2 \cdot 9) + 9 h$ .

$\frac{9 (2 \cdot 9 + h) + 9 h + {(9)}^{2}}{((9) + 9) ((9) + h + 9)}$

Step 6.4

Factor $9$ out of $9 h$ .

$\frac{9 (2 \cdot 9 + h) + 9 (h) + {(9)}^{2}}{((9) + 9) ((9) + h + 9)}$

Step 6.5

Factor $9$ out of $9 (2 \cdot 9 + h) + 9 (h)$ .

$\frac{9 (2 \cdot 9 + h + h) + {(9)}^{2}}{((9) + 9) ((9) + h + 9)}$

Step 6.6

Factor $9$ out of ${(9)}^{2}$ .

$\frac{9 (2 \cdot 9 + h + h) + 9 \cdot 9}{((9) + 9) ((9) + h + 9)}$

Step 6.7

Factor $9$ out of $9 (2 \cdot 9 + h + h) + 9 \cdot 9$ .

$\frac{9 (2 \cdot 9 + h + h + 9)}{((9) + 9) ((9) + h + 9)}$

Step 6.8

Cancel the common factors.

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Step 6.8.1

Factor $9$ out of $((9) + 9) ((9) + h + 9)$ .

$\frac{9 (2 \cdot 9 + h + h + 9)}{9 ((1 + 1) ((9) + h + 9))}$

Step 6.8.2

Cancel the common factor.

$\frac{9 (2 \cdot 9 + h + h + 9)}{9 ((1 + 1) ((9) + h + 9))}$

Step 6.8.3

Rewrite the expression.

$\frac{2 \cdot 9 + h + h + 9}{(1 + 1) ((9) + h + 9)}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $2$ by $9$ .

$\frac{18 + h + h + 9}{(1 + 1) (9 + h + 9)}$

Step 7.2

Add $18$ and $9$ .

$\frac{h + h + 27}{(1 + 1) (9 + h + 9)}$

Step 7.3

Add $h$ and $h$ .

$\frac{2 h + 27}{(1 + 1) (9 + h + 9)}$

Step 8

Simplify the denominator.

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Step 8.1

Add $1$ and $1$ .

$\frac{2 h + 27}{2 (9 + h + 9)}$

Step 8.2

Add $9$ and $9$ .

$\frac{2 h + 27}{2 (h + 18)}$

Step 9