Precalculus Examples

$f (x) = - 2 x - 3$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

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

$- 2 x - 2 h - 3$

$- 2 x - 2 h - 3$

$- 2 x - 2 h - 3$

Step 2.2

Reorder $- 2 x$ and $- 2 h$ .

$- 2 h - 2 x - 3$

Step 2.3

Find the components of the definition.

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

$f (x) = - 2 x - 3$

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

$f (x) = - 2 x - 3$

Step 3

Plug in the components.

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

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.

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

Step 4.1.2

Multiply $- 2$ by $- 1$ .

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

Step 4.1.3

Multiply $- 1$ by $- 3$ .

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

Step 4.1.4

Add $- 2 x$ and $2 x$ .

$\frac{- 2 h + 0 - 3 + 3}{h}$

Step 4.1.5

Add $- 2 h$ and $0$ .

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

Step 4.1.6

Add $- 3$ and $3$ .

$\frac{- 2 h + 0}{h}$

Step 4.1.7

Add $- 2 h$ and $0$ .

$\frac{- 2 h}{h}$

$\frac{- 2 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{- 2 h}{h}$

Step 4.2.2

Divide $- 2$ by $1$ .

$- 2$

$- 2$

$- 2$

Step 5

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