Precalculus Examples

$f (x) = - 2 x^{2} + 4$

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.

$f (x + h) = - 2 {(x + h)}^{2} + 4$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = - 2 ((x + h) (x + h)) + 4$

Step 2.1.2.1.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.1.2.1

Apply the distributive property.

$f (x + h) = - 2 (x (x + h) + h (x + h)) + 4$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = - 2 (x \cdot x + x h + h (x + h)) + 4$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = - 2 (x \cdot x + x h + h x + h \cdot h) + 4$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = - 2 (x^{2} + x h + h x + h \cdot h) + 4$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = - 2 (x^{2} + x h + h x + h^{2}) + 4$

Step 2.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = - 2 (x^{2} + h x + h x + h^{2}) + 4$

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = - 2 (x^{2} + 2 h x + h^{2}) + 4$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = - 2 x^{2} - 2 (2 h x) - 2 h^{2} + 4$

Step 2.1.2.1.5

Multiply $2$ by $- 2$ .

$f (x + h) = - 2 x^{2} - 4 h x - 2 h^{2} + 4$

Step 2.1.2.2

The final answer is $- 2 x^{2} - 4 h x - 2 h^{2} + 4$ .

$- 2 x^{2} - 4 h x - 2 h^{2} + 4$

Step 2.2

Reorder.

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Step 2.2.1

Move $- 2 x^{2}$ .

$- 4 h x - 2 h^{2} - 2 x^{2} + 4$

Step 2.2.2

Reorder $- 4 h x$ and $- 2 h^{2}$ .

$- 2 h^{2} - 4 h x - 2 x^{2} + 4$

Step 2.3

Find the components of the definition.

$f (x + h) = - 2 h^{2} - 4 h x - 2 x^{2} + 4$

$f (x) = - 2 x^{2} + 4$

$f (x + h) = - 2 h^{2} - 4 h x - 2 x^{2} + 4$

$f (x) = - 2 x^{2} + 4$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 2 h^{2} - 4 h x - 2 x^{2} + 4 - (- 2 x^{2} + 4)}{h}$

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.

$\frac{- 2 h^{2} - 4 h x - 2 x^{2} + 4 - (- 2 x^{2}) - 1 \cdot 4}{h}$

Step 4.1.2

Multiply $- 2$ by $- 1$ .

$\frac{- 2 h^{2} - 4 h x - 2 x^{2} + 4 + 2 x^{2} - 1 \cdot 4}{h}$

Step 4.1.3

Multiply $- 1$ by $4$ .

$\frac{- 2 h^{2} - 4 h x - 2 x^{2} + 4 + 2 x^{2} - 4}{h}$

Step 4.1.4

Add $- 2 x^{2}$ and $2 x^{2}$ .

$\frac{- 2 h^{2} - 4 h x + 0 + 4 - 4}{h}$

Step 4.1.5

Add $- 2 h^{2}$ and $0$ .

$\frac{- 2 h^{2} - 4 h x + 4 - 4}{h}$

Step 4.1.6

Subtract $4$ from $4$ .

$\frac{- 2 h^{2} - 4 h x + 0}{h}$

Step 4.1.7

Add $- 2 h^{2} - 4 h x$ and $0$ .

$\frac{- 2 h^{2} - 4 h x}{h}$

Step 4.1.8

Factor $2 h$ out of $- 2 h^{2} - 4 h x$ .

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Step 4.1.8.1

Factor $2 h$ out of $- 2 h^{2}$ .

$\frac{2 h (- h) - 4 h x}{h}$

Step 4.1.8.2

Factor $2 h$ out of $- 4 h x$ .

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

Step 4.1.8.3

Factor $2 h$ out of $2 h (- h) + 2 h (- 2 x)$ .

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

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $2 (- h - 2 x)$ by $1$ .

$2 (- h - 2 x)$

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Multiply $- 1$ by $2$ .

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

Step 4.2.3.2

Multiply $- 2$ by $2$ .

$- 2 h - 4 x$

Step 4.2.3.3

Reorder $- 2 h$ and $- 4 x$ .

$- 4 x - 2 h$

Step 5