Examples

f (x) = x^{2}

$f (x) = x^{2}$ ,

x = 0

$x = 0$

Step 1

Consider the difference quotient formula.

\frac{f (x + h) - f (x)}{h}

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at

x = x + h

$x = x + h$ .

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Step 2.1.1

Replace the variable

x

$x$ with

x + h

$x + h$ in the expression.

f (x + h) = {(x + h)}^{2}

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Rewrite

{(x + h)}^{2}

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

(x + h) (x + h)

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

f (x + h) = (x + h) (x + h)

$f (x + h) = (x + h) (x + h)$

Step 2.1.2.2

Expand

(x + h) (x + h)

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

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Step 2.1.2.2.1

Apply the distributive property.

f (x + h) = x (x + h) + h (x + h)

$f (x + h) = x (x + h) + h (x + h)$

Step 2.1.2.2.2

Apply the distributive property.

f (x + h) = x \cdot x + x h + h (x + h)

$f (x + h) = x \cdot x + x h + h (x + h)$

Step 2.1.2.2.3

Apply the distributive property.

f (x + h) = x \cdot x + x h + h x + h \cdot h

$f (x + h) = x \cdot x + x h + h x + h \cdot h$

f (x + h) = x \cdot x + x h + h x + h \cdot h

$f (x + h) = x \cdot x + x h + h x + h \cdot h$

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

$x$ by

x

$x$ .

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

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

Step 2.1.2.3.1.2

Multiply

h

$h$ by

h

$h$ .

f (x + h) = x^{2} + x h + h x + h^{2}

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

f (x + h) = x^{2} + x h + h x + h^{2}

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

Step 2.1.2.3.2

Add

x h

$x h$ and

h x

$h x$ .

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Step 2.1.2.3.2.1

Reorder

x

$x$ and

h

$h$ .

f (x + h) = x^{2} + h x + h x + h^{2}

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

Step 2.1.2.3.2.2

Add

h x

$h x$ and

h x

$h x$ .

f (x + h) = x^{2} + 2 h x + h^{2}

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

f (x + h) = x^{2} + 2 h x + h^{2}

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

f (x + h) = x^{2} + 2 h x + h^{2}

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

Step 2.1.2.4

The final answer is

x^{2} + 2 h x + h^{2}

$x^{2} + 2 h x + h^{2}$ .

x^{2} + 2 h x + h^{2}

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

x^{2} + 2 h x + h^{2}

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

x^{2} + 2 h x + h^{2}

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

Step 2.2

Reorder.

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Step 2.2.1

Move

x^{2}

$x^{2}$ .

2 h x + h^{2} + x^{2}

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

Step 2.2.2

Reorder

2 h x

$2 h x$ and

h^{2}

$h^{2}$ .

h^{2} + 2 h x + x^{2}

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

h^{2} + 2 h x + x^{2}

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

Step 2.3

Find the components of the definition.

f (x + h) = h^{2} + 2 h x + x^{2}

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

f (x) = x^{2}

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

f (x + h) = h^{2} + 2 h x + x^{2}

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

f (x) = x^{2}

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

Step 3

Plug in the components.

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Subtract

x^{2}

$x^{2}$ from

x^{2}

$x^{2}$ .

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

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

Step 4.1.2

Add

h^{2} + 2 h x

$h^{2} + 2 h x$ and

0

$0$ .

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

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

Step 4.1.3

Factor

h

$h$ out of

h^{2} + 2 h x

$h^{2} + 2 h x$ .

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Step 4.1.3.1

Factor

h

$h$ out of

h^{2}

$h^{2}$ .

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

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

Step 4.1.3.2

Factor

h

$h$ out of

2 h x

$2 h x$ .

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

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

Step 4.1.3.3

Factor

h

$h$ out of

h (h) + h (2 x)

$h (h) + h (2 x)$ .

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

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

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

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

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

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

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Cancel the common factor of

h

$h$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide

$h + 2 x$ by

$1$ .

$h + 2 x$

Step 4.2.2

Reorder

$h$ and

$2 x$ .

$2 x + h$

Step 5

Replace the variable

$x$ with

$0$ in the expression.

$2 (0) + h$

Step 6

Multiply

$2$ by

$0$ .

$0 + h$

Step 7

Add

$0$ and

$h$ .

$h$

Step 8

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