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Examples

$f (x) = 8 x - 3$ , $x = 1$

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.

$f (x + h) = 8 (x + h) - 3$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = 8 x + 8 h - 3$

Step 2.1.2.2

The final answer is $8 x + 8 h - 3$ .

$8 x + 8 h - 3$

$8 x + 8 h - 3$

$8 x + 8 h - 3$

Step 2.2

Reorder $8 x$ and $8 h$ .

$8 h + 8 x - 3$

Step 2.3

Find the components of the definition.

$f (x + h) = 8 h + 8 x - 3$

$f (x) = 8 x - 3$

$f (x + h) = 8 h + 8 x - 3$

$f (x) = 8 x - 3$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{8 h + 8 x - 3 - (8 x - 3)}{h}$

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.

$\frac{8 h + 8 x - 3 - (8 x) - - 3}{h}$

Step 4.1.2

Multiply $8$ by $- 1$ .

$\frac{8 h + 8 x - 3 - 8 x - - 3}{h}$

Step 4.1.3

Multiply $- 1$ by $- 3$ .

$\frac{8 h + 8 x - 3 - 8 x + 3}{h}$

Step 4.1.4

Subtract $8 x$ from $8 x$ .

$\frac{8 h + 0 - 3 + 3}{h}$

Step 4.1.5

Add $8 h$ and $0$ .

$\frac{8 h - 3 + 3}{h}$

Step 4.1.6

Add $- 3$ and $3$ .

$\frac{8 h + 0}{h}$

Step 4.1.7

Add $8 h$ and $0$ .

$\frac{8 h}{h}$

$\frac{8 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{8 h}{h}$

Step 4.2.2

Divide $8$ by $1$ .

$8$

$8$

$8$

Step 5

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$