Precalculus Examples

$2 x + h$

Step 1

Write $2 x + h$ as a function.

$f (x) = 2 x + h$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = x + h$ .

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Step 3.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 2 (x + h) + h$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (x + h) = 2 x + 2 h + h$

Step 3.1.2.2

Add $2 h$ and $h$ .

$f (x + h) = 2 x + 3 h$

Step 3.1.2.3

The final answer is $2 x + 3 h$ .

$2 x + 3 h$

Step 3.2

Find the components of the definition.

$f (x + h) = 2 x + 3 h$

$f (x) = 2 x + h$

$f (x + h) = 2 x + 3 h$

$f (x) = 2 x + h$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{2 x + 3 h - (2 x + h)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{2 x + 3 h - (2 x) - h}{h}$

Step 5.1.2

Multiply $2$ by $- 1$ .

$\frac{2 x + 3 h - 2 x - h}{h}$

Step 5.1.3

Subtract $2 x$ from $2 x$ .

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

Step 5.1.4

Add $3 h$ and $0$ .

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

Step 5.1.5

Subtract $h$ from $3 h$ .

$\frac{2 h}{h}$

Step 5.2

Cancel the common factor of $h$ .

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Step 5.2.1

Cancel the common factor.

$\frac{2 h}{h}$

Step 5.2.2

Divide $2$ by $1$ .

$2$

Step 6