Precalculus Examples

$f (x) = \frac{1}{2} x^{2} + 4 x$

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.

$f (x + h) = \frac{1}{2} \cdot {(x + h)}^{2} + 4 (x + h)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

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

$f (x + h) = \frac{1}{2} \cdot ((x + h) (x + h)) + 4 (x + h)$

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.

$f (x + h) = \frac{1}{2} \cdot (x (x + h) + h (x + h)) + 4 (x + h)$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = \frac{1}{2} \cdot (x \cdot x + x h + h (x + h)) + 4 (x + h)$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = \frac{1}{2} \cdot (x \cdot x + x h + h x + h \cdot h) + 4 (x + h)$

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 $x$ .

$f (x + h) = \frac{1}{2} \cdot (x^{2} + x h + h x + h \cdot h) + 4 (x + h)$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = \frac{1}{2} \cdot (x^{2} + x h + h x + h^{2}) + 4 (x + h)$

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 $h$ .

$f (x + h) = \frac{1}{2} \cdot (x^{2} + h x + h x + h^{2}) + 4 (x + h)$

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = \frac{1}{2} \cdot (x^{2} + 2 h x + h^{2}) + 4 (x + h)$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = \frac{1}{2} x^{2} + \frac{1}{2} \cdot (2 h x) + \frac{1}{2} h^{2} + 4 (x + h)$

Step 2.1.2.1.5

Simplify.

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Step 2.1.2.1.5.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$f (x + h) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (2 h x) + \frac{1}{2} h^{2} + 4 (x + h)$

Step 2.1.2.1.5.2

Cancel the common factor of $2$ .

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Step 2.1.2.1.5.2.1

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

$f (x + h) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (2 (h x)) + \frac{1}{2} h^{2} + 4 (x + h)$

Step 2.1.2.1.5.2.2

Cancel the common factor.

$f (x + h) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (2 (h x)) + \frac{1}{2} h^{2} + 4 (x + h)$

Step 2.1.2.1.5.2.3

Rewrite the expression.

$f (x + h) = \frac{x^{2}}{2} + h x + \frac{1}{2} h^{2} + 4 (x + h)$

Step 2.1.2.1.5.3

Combine $\frac{1}{2}$ and $h^{2}$ .

$f (x + h) = \frac{x^{2}}{2} + h x + \frac{h^{2}}{2} + 4 (x + h)$

Step 2.1.2.1.6

Apply the distributive property.

$f (x + h) = \frac{x^{2}}{2} + h x + \frac{h^{2}}{2} + 4 x + 4 h$

Step 2.1.2.2

The final answer is $\frac{x^{2}}{2} + h x + \frac{h^{2}}{2} + 4 x + 4 h$ .

$\frac{x^{2}}{2} + h x + \frac{h^{2}}{2} + 4 x + 4 h$

Step 2.2

Reorder.

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Step 2.2.1

Move $4 x$ .

$\frac{x^{2}}{2} + h x + \frac{h^{2}}{2} + 4 h + 4 x$

Step 2.2.2

Move $\frac{x^{2}}{2}$ .

$h x + \frac{h^{2}}{2} + \frac{x^{2}}{2} + 4 h + 4 x$

Step 2.2.3

Reorder $h x$ and $\frac{h^{2}}{2}$ .

$\frac{h^{2}}{2} + h x + \frac{x^{2}}{2} + 4 h + 4 x$

Step 2.3

Find the components of the definition.

$f (x + h) = \frac{h^{2}}{2} + h x + \frac{x^{2}}{2} + 4 h + 4 x$

$f (x) = \frac{x^{2}}{2} + 4 x$

$f (x + h) = \frac{h^{2}}{2} + h x + \frac{x^{2}}{2} + 4 h + 4 x$

$f (x) = \frac{x^{2}}{2} + 4 x$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{h^{2}}{2} + h x + \frac{x^{2}}{2} + 4 h + 4 x - (\frac{x^{2}}{2} + 4 x)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{\frac{h^{2}}{2} + h x + \frac{x^{2}}{2} + 4 h + 4 x - \frac{x^{2}}{2} - (4 x)}{h}$

Step 4.1.2

Multiply $4$ by $- 1$ .

$\frac{\frac{h^{2}}{2} + h x + \frac{x^{2}}{2} + 4 h + 4 x - \frac{x^{2}}{2} - 4 x}{h}$

Step 4.1.3

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

$\frac{\frac{h^{2}}{2} + h x + 4 h + 4 x + 0 - 4 x}{h}$

Step 4.1.4

Add $\frac{h^{2}}{2}$ and $0$ .

$\frac{\frac{h^{2}}{2} + h x + 4 h + 4 x - 4 x}{h}$

Step 4.1.5

Subtract $4 x$ from $4 x$ .

$\frac{\frac{h^{2}}{2} + h x + 4 h + 0}{h}$

Step 4.1.6

Add $\frac{h^{2}}{2} + h x + 4 h$ and $0$ .

$\frac{\frac{h^{2}}{2} + h x + 4 h}{h}$

Step 4.1.7

Factor $h$ out of $\frac{h^{2}}{2} + h x + 4 h$ .

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Step 4.1.7.1

Factor $h$ out of $\frac{h^{2}}{2}$ .

$\frac{h \frac{h}{2} + h x + 4 h}{h}$

Step 4.1.7.2

Factor $h$ out of $h x$ .

$\frac{h (\frac{h}{2}) + h (x) + 4 h}{h}$

Step 4.1.7.3

Factor $h$ out of $4 h$ .

$\frac{h (\frac{h}{2}) + h (x) + h \cdot 4}{h}$

Step 4.1.7.4

Factor $h$ out of $h (\frac{h}{2}) + h (x)$ .

$\frac{h (\frac{h}{2} + x) + h \cdot 4}{h}$

Step 4.1.7.5

Factor $h$ out of $h (\frac{h}{2} + x) + h \cdot 4$ .

$\frac{h (\frac{h}{2} + x + 4)}{h}$

Step 4.1.8

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

$\frac{h (\frac{h}{2} + x \cdot \frac{2}{2} + 4)}{h}$

Step 4.1.9

Combine $x$ and $\frac{2}{2}$ .

$\frac{h (\frac{h}{2} + \frac{x \cdot 2}{2} + 4)}{h}$

Step 4.1.10

Combine the numerators over the common denominator.

$\frac{h (\frac{h + x \cdot 2}{2} + 4)}{h}$

Step 4.1.11

Move $2$ to the left of $x$ .

$\frac{h (\frac{h + 2 x}{2} + 4)}{h}$

Step 4.1.12

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{h (\frac{h + 2 x}{2} + 4 \cdot \frac{2}{2})}{h}$

Step 4.1.13

Combine $4$ and $\frac{2}{2}$ .

$\frac{h (\frac{h + 2 x}{2} + \frac{4 \cdot 2}{2})}{h}$

Step 4.1.14

Combine the numerators over the common denominator.

$\frac{h \frac{h + 2 x + 4 \cdot 2}{2}}{h}$

Step 4.1.15

Multiply $4$ by $2$ .

$\frac{h \frac{h + 2 x + 8}{2}}{h}$

Step 4.2

Combine $h$ and $\frac{h + 2 x + 8}{2}$ .

$\frac{\frac{h (h + 2 x + 8)}{2}}{h}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{h (h + 2 x + 8)}{2} \cdot \frac{1}{h}$

Step 4.4

Cancel the common factor of $h$ .

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Step 4.4.1

Cancel the common factor.

$\frac{h (h + 2 x + 8)}{2} \cdot \frac{1}{h}$

Step 4.4.2

Rewrite the expression.

$\frac{h + 2 x + 8}{2}$

Step 5