Calculus Examples

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

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.

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

Step 2.1.2

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

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Combine the numerators over the common denominator.

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

Step 4.1.2

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

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Step 4.1.2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + h$ and $b = x$ .

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

Step 4.1.2.2

Simplify.

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Step 4.1.2.2.1

Add $x$ and $x$ .

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

Step 4.1.2.2.2

Subtract $x$ from $x$ .

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

Step 4.1.2.2.3

Add $h$ and $0$ .

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factor.

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

Step 4.3.3

Rewrite the expression.

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

Step 5