Precalculus Examples

$\frac{f (x + h) - f (x)}{h}$ , $f (x) = 2 x - 1$

Step 1

Evaluate the numerator in $\frac{f (x + h) - f (x)}{h}$ .

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Step 1.1

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

$f (x + h) = 2 (x + h) - 1$

Step 1.2

Replace the function designators in $f (x + h) - f (x)$ with the actual functions.

$(2 (x + h) - 1) - (2 x - 1)$

Step 1.3

Simplify $(2 (x + h) - 1) - (2 x - 1)$ .

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Apply the distributive property.

$2 x + 2 h - 1 - (2 x - 1)$

Step 1.3.1.2

Apply the distributive property.

$2 x + 2 h - 1 - (2 x) - - 1$

Step 1.3.1.3

Multiply $2$ by $- 1$ .

$2 x + 2 h - 1 - 2 x - - 1$

Step 1.3.1.4

Multiply $- 1$ by $- 1$ .

$2 x + 2 h - 1 - 2 x + 1$

Step 1.3.2

Combine the opposite terms in $2 x + 2 h - 1 - 2 x + 1$ .

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Step 1.3.2.1

Subtract $2 x$ from $2 x$ .

$2 h - 1 + 0 + 1$

Step 1.3.2.2

Add $2 h - 1$ and $0$ .

$2 h - 1 + 1$

Step 1.3.2.3

Add $- 1$ and $1$ .

$2 h + 0$

Step 1.3.2.4

Add $2 h$ and $0$ .

$2 h$

Step 2

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

$\frac{2 h}{h}$

Step 3

Cancel the common factor of $h$ .

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Step 3.1

Cancel the common factor.

$\frac{2 h}{h}$

Step 3.2

Divide $2$ by $1$ .

$2$