Precalculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the denominator.

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Step 2.1.2.1.1

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

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

Step 2.1.2.1.2

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

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Step 2.1.2.1.2.1

Apply the distributive property.

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

Step 2.1.2.1.2.2

Apply the distributive property.

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

Step 2.1.2.1.2.3

Apply the distributive property.

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

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

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

Step 2.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.3.2.1

Reorder $x$ and $h$ .

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

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.2

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

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Rewrite $\frac{3 (x^{2} + 2) - 3 (x^{2} + 2 h x + h^{2} + 2)}{(x^{2} + 2) (x^{2} + 2 h x + h^{2} + 2)}$ in a factored form.

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Step 4.1.5.1

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

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Step 4.1.5.1.1

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

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

Step 4.1.5.1.2

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

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

Step 4.1.5.2

Apply the distributive property.

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

Step 4.1.5.3

Simplify.

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Step 4.1.5.3.1

Multiply $2$ by $- 1$ .

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

Step 4.1.5.3.2

Multiply $- 1$ by $2$ .

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

Step 4.1.5.4

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

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

Step 4.1.5.5

Add $0$ and $2$ .

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

Step 4.1.5.6

Subtract $2$ from $2$ .

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

Step 4.1.5.7

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

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

Step 4.1.5.8

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

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Step 4.1.5.8.1

Factor $h$ out of $- 2 h x$ .

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

Step 4.1.5.8.2

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

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

Step 4.1.5.8.3

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

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

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Combine.

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

Step 4.4

Cancel the common factor of $h$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Multiply $3$ by $1$ .

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

Step 4.6

Factor $- 1$ out of $- 2 x$ .

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

Step 4.7

Factor $- 1$ out of $- (2 x) - h$ .

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

Step 4.8

Simplify the expression.

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Step 4.8.1

Rewrite $- (2 x + h)$ as $- 1 (2 x + h)$ .

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

Step 4.8.2

Move the negative in front of the fraction.

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

Step 5