Calculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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Step 2.1.2.2.1

Apply the distributive property.

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

Step 2.1.2.2.2

Apply the distributive property.

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

Step 2.1.2.2.3

Apply the distributive property.

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

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.2.3.1.2

Multiply $h$ by $h$ .

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

Step 2.1.2.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.3.2.1

Reorder $x$ and $h$ .

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

Step 2.1.2.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.4

Apply the distributive property.

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

Step 2.1.2.5

Simplify.

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Step 2.1.2.5.1

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

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

Step 2.1.2.5.2

Cancel the common factor of $2$ .

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Step 2.1.2.5.2.1

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

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

Step 2.1.2.5.2.2

Cancel the common factor.

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

Step 2.1.2.5.2.3

Rewrite the expression.

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

Step 2.1.2.5.3

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

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

Step 2.1.2.6

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

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

Step 2.2

Reorder.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

Factor $h$ out of $h x$ .

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

Step 4.1.3.3

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

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.1.6

Combine the numerators over the common denominator.

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

Step 4.1.7

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

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

Step 4.2

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

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

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.4

Cancel the common factor of $h$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 5