Examples

f (x) = 4 x - 4

$f (x) = 4 x - 4$ ,

x = 2

$x = 2$

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) = 4 (x + h) - 4

$f (x + h) = 4 (x + h) - 4$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

f (x + h) = 4 x + 4 h - 4

$f (x + h) = 4 x + 4 h - 4$

Step 2.1.2.2

The final answer is

4 x + 4 h - 4

$4 x + 4 h - 4$ .

4 x + 4 h - 4

$4 x + 4 h - 4$

4 x + 4 h - 4

$4 x + 4 h - 4$

4 x + 4 h - 4

$4 x + 4 h - 4$

Step 2.2

Reorder

4 x

$4 x$ and

4 h

$4 h$ .

4 h + 4 x - 4

$4 h + 4 x - 4$

Step 2.3

Find the components of the definition.

f (x + h) = 4 h + 4 x - 4

$f (x + h) = 4 h + 4 x - 4$

f (x) = 4 x - 4

$f (x) = 4 x - 4$

f (x + h) = 4 h + 4 x - 4

$f (x + h) = 4 h + 4 x - 4$

f (x) = 4 x - 4

$f (x) = 4 x - 4$

Step 3

Plug in the components.

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor

4

$4$ out of

4 x - 4

$4 x - 4$ .

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Step 4.1.1.1

Factor

4

$4$ out of

4 x

$4 x$ .

\frac{4 h + 4 x - 4 - (4 (x) - 4)}{h}

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

Step 4.1.1.2

Factor

4

$4$ out of

- 4

$- 4$ .

\frac{4 h + 4 x - 4 - (4 (x) + 4 (- 1))}{h}

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

Step 4.1.1.3

Factor

4

$4$ out of

4 (x) + 4 (- 1)

$4 (x) + 4 (- 1)$ .

\frac{4 h + 4 x - 4 - (4 (x - 1))}{h}

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

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

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

Step 4.1.2

Multiply

- 1

$- 1$ by

4

$4$ .

\frac{4 h + 4 x - 4 - 4 (x - 1)}{h}

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

Step 4.1.3

Factor

4

$4$ out of

4 h + 4 x - 4 - 4 (x - 1)

$4 h + 4 x - 4 - 4 (x - 1)$ .

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Step 4.1.3.1

Factor

4

$4$ out of

4 h

$4 h$ .

\frac{4 h + 4 x - 4 - 4 (x - 1)}{h}

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

Step 4.1.3.2

Factor

4

$4$ out of

4 x

$4 x$ .

\frac{4 h + 4 (x) - 4 - 4 (x - 1)}{h}

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

Step 4.1.3.3

Factor

4

$4$ out of

- 4

$- 4$ .

\frac{4 h + 4 (x) + 4 (- 1) - 4 (x - 1)}{h}

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

Step 4.1.3.4

Factor

4

$4$ out of

- 4 (x - 1)

$- 4 (x - 1)$ .

\frac{4 h + 4 (x) + 4 (- 1) + 4 (- (x - 1))}{h}

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

Step 4.1.3.5

Factor

4

$4$ out of

4 h + 4 (x)

$4 h + 4 (x)$ .

\frac{4 (h + x) + 4 (- 1) + 4 (- (x - 1))}{h}

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

Step 4.1.3.6

Factor

4

$4$ out of

4 (h + x) + 4 (- 1)

$4 (h + x) + 4 (- 1)$ .

\frac{4 (h + x - 1) + 4 (- (x - 1))}{h}

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

Step 4.1.3.7

Factor

4

$4$ out of

4 (h + x - 1) + 4 (- (x - 1))

$4 (h + x - 1) + 4 (- (x - 1))$ .

\frac{4 (h + x - 1 - (x - 1))}{h}

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

\frac{4 (h + x - 1 - (x - 1))}{h}

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

Step 4.1.4

Apply the distributive property.

\frac{4 (h + x - 1 - x - - 1)}{h}

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

Step 4.1.5

Multiply

- 1

$- 1$ by

$- 1$ .

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

Step 4.1.6

Subtract

$x$ from

$x$ .

$\frac{4 (h + 0 - 1 + 1)}{h}$

Step 4.1.7

Add

$h$ and

$0$ .

$\frac{4 (h - 1 + 1)}{h}$

Step 4.1.8

Add

$- 1$ and

$1$ .

$\frac{4 (h + 0)}{h}$

Step 4.1.9

Add

$h$ and

$0$ .

$\frac{4 h}{h}$

Step 4.2

Cancel the common factor of

$h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{4 h}{h}$

Step 4.2.2

Divide

$4$ by

$1$ .

$4$

Step 5

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