Precalculus Examples

$f (x) = \frac{3}{x^{3}}$

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.

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

Step 2.1.2

The final answer is $\frac{3}{{(x + h)}^{3}}$ .

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

Step 2.2

Find the components of the definition.

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

$f (x) = \frac{3}{x^{3}}$

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

$f (x) = \frac{3}{x^{3}}$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Factor $3$ out of $3 x^{3} - 3 {(x + h)}^{3}$ .

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Step 4.1.5.1.1

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

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

Step 4.1.5.1.2

Factor $3$ out of $- 3 {(x + h)}^{3}$ .

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

Step 4.1.5.1.3

Factor $3$ out of $3 (x^{3}) + 3 (- {(x + h)}^{3})$ .

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

Step 4.1.5.2

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 $a = x$ and $b = x + h$ .

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

Step 4.1.5.3

Simplify.

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Step 4.1.5.3.1

Apply the distributive property.

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

Step 4.1.5.3.2

Subtract $x$ from $x$ .

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

Step 4.1.5.3.3

Subtract $h$ from $0$ .

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

Step 4.1.5.3.4

Apply the distributive property.

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

Step 4.1.5.3.5

Multiply $x$ by $x$ .

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

Step 4.1.5.3.6

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

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

Step 4.1.5.3.7

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

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Step 4.1.5.3.7.1

Apply the distributive property.

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

Step 4.1.5.3.7.2

Apply the distributive property.

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

Step 4.1.5.3.7.3

Apply the distributive property.

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

Step 4.1.5.3.8

Simplify and combine like terms.

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Step 4.1.5.3.8.1

Simplify each term.

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Step 4.1.5.3.8.1.1

Multiply $x$ by $x$ .

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

Step 4.1.5.3.8.1.2

Multiply $h$ by $h$ .

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

Step 4.1.5.3.8.2

Add $x h$ and $h x$ .

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Step 4.1.5.3.8.2.1

Reorder $h$ and $x$ .

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

Step 4.1.5.3.8.2.2

Add $x h$ and $x h$ .

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

Step 4.1.5.3.9

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

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

Step 4.1.5.3.10

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

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

Step 4.1.5.3.11

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

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

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

Step 4.1.5.4

Combine exponents.

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Step 4.1.5.4.1

Factor out negative.

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

Step 4.1.5.4.2

Multiply $3$ by $- 1$ .

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

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

Step 4.1.6

Move the negative in front of the fraction.

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{3 h (3 x^{2} + 3 x h + h^{2})}{x^{3} {(x + h)}^{3}}$ into the numerator.

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

Step 4.3.2

Factor $h$ out of $- 3 h (3 x^{2} + 3 x h + h^{2})$ .

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5