Precalculus Examples

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

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.

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

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 $(x + h) (x + h)$ .

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

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.

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

Step 2.1.2.1.2.2

Apply the distributive property.

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

Step 2.1.2.1.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

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

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 $h$ .

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

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.1.4

Apply the distributive property.

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

Step 2.1.2.1.5

Simplify.

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Step 2.1.2.1.5.1

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

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

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 $2 h x$ .

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

Step 2.1.2.1.5.2.2

Cancel the common factor.

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

Step 2.1.2.1.5.2.3

Rewrite the expression.

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

Step 2.1.2.1.5.3

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

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

Step 2.1.2.1.6

Apply the distributive property.

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

Step 2.1.2.1.7

Combine $\frac{1}{4}$ and $x$ .

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

Step 2.1.2.1.8

Combine $\frac{1}{4}$ and $h$ .

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

Step 2.1.2.2

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

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

Step 2.2

Reorder.

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Step 2.2.1

Move $\frac{x}{4}$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

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.

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

Step 4.1.2

Reorder the factors of $h x$ .

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

Step 4.1.3

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

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

Step 4.1.4

Combine $x h$ and $\frac{4}{4}$ .

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

Step 4.1.5

Combine the numerators over the common denominator.

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

Step 4.1.6

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

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

Step 4.1.7

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.7.1

Multiply $\frac{h^{2}}{2}$ by $\frac{2}{2}$ .

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

Step 4.1.7.2

Multiply $2$ by $2$ .

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

Step 4.1.8

Combine the numerators over the common denominator.

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

Step 4.1.9

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

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

Step 4.1.10

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.10.1

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

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

Step 4.1.10.2

Multiply $2$ by $2$ .

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

Step 4.1.11

Combine the numerators over the common denominator.

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

Step 4.1.12

Combine the numerators over the common denominator.

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

Step 4.1.13

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

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

Step 4.1.14

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.14.1

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

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

Step 4.1.14.2

Multiply $2$ by $2$ .

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

Step 4.1.15

Combine the numerators over the common denominator.

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

Step 4.1.16

Combine the numerators over the common denominator.

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

Step 4.1.17

Rewrite $\frac{x^{2} \cdot 2 + h^{2} \cdot 2 + x h \cdot 4 + h + x - x^{2} \cdot 2 - x}{4}$ in a factored form.

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Step 4.1.17.1

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

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

Step 4.1.17.2

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

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

Step 4.1.17.3

Subtract $x$ from $x$ .

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

Step 4.1.17.4

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

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

Step 4.1.17.5

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

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Step 4.1.17.5.1

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

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

Step 4.1.17.5.2

Factor $h$ out of $x h \cdot 4$ .

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

Step 4.1.17.5.3

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

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

Step 4.1.17.5.4

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

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

Step 4.1.17.5.5

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

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

Step 4.1.17.5.6

Factor $h$ out of $h (h \cdot 2 + x \cdot 4) + h \cdot 1$ .

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

Step 4.1.17.6

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

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

Step 4.1.17.7

Move $4$ to the left of $x$ .

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$\frac{2 h + 4 x + 1}{4}$

Step 5