Precalculus Examples

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

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.

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

Step 2.1.2

The final answer is $\frac{3}{{(x + h)}^{2}}$ .

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

Step 2.2

Find the components of the definition.

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

$f (x) = \frac{3}{x^{2}}$

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

$f (x) = \frac{3}{x^{2}}$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{3}{{(x + h)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

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

Step 4.1.2

To write $- \frac{3}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x + h)}^{2}}{{(x + h)}^{2}}$ .

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

Step 4.1.3

Write each expression with a common denominator of ${(x + h)}^{2} x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

Reorder the factors of ${(x + h)}^{2} x^{2}$ .

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

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

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Step 4.1.5.1.1

Factor $3$ out of $3 x^{2}$ .

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

Step 4.1.5.1.2

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

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

Step 4.1.5.1.3

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

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

Step 4.1.5.2

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 $b = x + h$ .

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

Step 4.1.5.3

Simplify.

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Step 4.1.5.3.1

Add $x$ and $x$ .

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

Step 4.1.5.3.2

Apply the distributive property.

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

Step 4.1.5.3.3

Subtract $x$ from $x$ .

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

Step 4.1.5.3.4

Subtract $h$ from $0$ .

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

Step 4.1.5.3.5

Factor out negative.

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

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

Step 4.1.5.4

Combine exponents.

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Step 4.1.5.4.1

Factor out negative.

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

Step 4.1.5.4.2

Multiply $3$ by $- 1$ .

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

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

Step 4.1.6

Move the negative in front of the fraction.

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5