Precalculus Examples

$f (x) = 4 x^{3}$

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.

$f (x + h) = 4 {(x + h)}^{3}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Use the Binomial Theorem.

$f (x + h) = 4 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3})$

Step 2.1.2.2

Apply the distributive property.

$f (x + h) = 4 x^{3} + 4 (3 x^{2} h) + 4 (3 x h^{2}) + 4 h^{3}$

Step 2.1.2.3

Simplify.

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Step 2.1.2.3.1

Multiply $3$ by $4$ .

$f (x + h) = 4 x^{3} + 12 (x^{2} h) + 4 (3 x h^{2}) + 4 h^{3}$

Step 2.1.2.3.2

Multiply $3$ by $4$ .

$f (x + h) = 4 x^{3} + 12 (x^{2} h) + 12 (x h^{2}) + 4 h^{3}$

Step 2.1.2.4

Remove parentheses.

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3}$

Step 2.1.2.5

The final answer is $4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3}$ .

$4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3}$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

$4 x^{3} + 12 h x^{2} + 12 x h^{2} + 4 h^{3}$

Step 2.2.2

Move $x$ .

$4 x^{3} + 12 h x^{2} + 12 h^{2} x + 4 h^{3}$

Step 2.2.3

Move $4 x^{3}$ .

$12 h x^{2} + 12 h^{2} x + 4 h^{3} + 4 x^{3}$

Step 2.2.4

Move $12 h x^{2}$ .

$12 h^{2} x + 4 h^{3} + 12 h x^{2} + 4 x^{3}$

Step 2.2.5

Reorder $12 h^{2} x$ and $4 h^{3}$ .

$4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3}$

Step 2.3

Find the components of the definition.

$f (x + h) = 4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3}$

$f (x) = 4 x^{3}$

$f (x + h) = 4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3}$

$f (x) = 4 x^{3}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} - (4 x^{3})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $4$ .

$\frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} - 4 x^{3}}{h}$

Step 4.1.2

Subtract $4 x^{3}$ from $4 x^{3}$ .

$\frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 0}{h}$

Step 4.1.3

Add $4 h^{3} + 12 h^{2} x + 12 h x^{2}$ and $0$ .

$\frac{4 h^{3} + 12 h^{2} x + 12 h x^{2}}{h}$

Step 4.1.4

Factor $4 h$ out of $4 h^{3} + 12 h^{2} x + 12 h x^{2}$ .

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Step 4.1.4.1

Factor $4 h$ out of $4 h^{3}$ .

$\frac{4 h \cdot h^{2} + 12 h^{2} x + 12 h x^{2}}{h}$

Step 4.1.4.2

Factor $4 h$ out of $12 h^{2} x$ .

$\frac{4 h \cdot h^{2} + 4 h (3 h x) + 12 h x^{2}}{h}$

Step 4.1.4.3

Factor $4 h$ out of $12 h x^{2}$ .

$\frac{4 h \cdot h^{2} + 4 h (3 h x) + 4 h (3 x^{2})}{h}$

Step 4.1.4.4

Factor $4 h$ out of $4 h \cdot h^{2} + 4 h (3 h x)$ .

$\frac{4 h (h^{2} + 3 h x) + 4 h (3 x^{2})}{h}$

Step 4.1.4.5

Factor $4 h$ out of $4 h (h^{2} + 3 h x) + 4 h (3 x^{2})$ .

$\frac{4 h (h^{2} + 3 h x + 3 x^{2})}{h}$

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.

$\frac{4 h (h^{2} + 3 h x + 3 x^{2})}{h}$

Step 4.2.1.2

Divide $4 (h^{2} + 3 h x + 3 x^{2})$ by $1$ .

$4 (h^{2} + 3 h x + 3 x^{2})$

Step 4.2.2

Apply the distributive property.

$4 h^{2} + 4 (3 h x) + 4 (3 x^{2})$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $3$ by $4$ .

$4 h^{2} + 12 (h x) + 4 (3 x^{2})$

Step 4.3.2

Multiply $3$ by $4$ .

$4 h^{2} + 12 (h x) + 12 x^{2}$

Step 4.4

Move $h$ .

$4 h^{2} + 12 x h + 12 x^{2}$

Step 4.5

Move $4 h^{2}$ .

$12 x h + 12 x^{2} + 4 h^{2}$

Step 4.6

Reorder $12 x h$ and $12 x^{2}$ .

$12 x^{2} + 12 x h + 4 h^{2}$

Step 5