Precalculus Examples

$2 x^{3}$

Step 1

Write $2 x^{3}$ as a function.

$f (x) = 2 x^{3}$

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.

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Use the Binomial Theorem.

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

Step 3.1.2.2

Apply the distributive property.

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

Step 3.1.2.3

Simplify.

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Step 3.1.2.3.1

Multiply $3$ by $2$ .

$f (x + h) = 2 x^{3} + 6 (x^{2} h) + 2 (3 x h^{2}) + 2 h^{3}$

Step 3.1.2.3.2

Multiply $3$ by $2$ .

$f (x + h) = 2 x^{3} + 6 (x^{2} h) + 6 (x h^{2}) + 2 h^{3}$

Step 3.1.2.4

Remove parentheses.

$f (x + h) = 2 x^{3} + 6 x^{2} h + 6 x h^{2} + 2 h^{3}$

Step 3.1.2.5

The final answer is $2 x^{3} + 6 x^{2} h + 6 x h^{2} + 2 h^{3}$ .

$2 x^{3} + 6 x^{2} h + 6 x h^{2} + 2 h^{3}$

Step 3.2

Reorder.

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Step 3.2.1

Move $x^{2}$ .

$2 x^{3} + 6 h x^{2} + 6 x h^{2} + 2 h^{3}$

Step 3.2.2

Move $x$ .

$2 x^{3} + 6 h x^{2} + 6 h^{2} x + 2 h^{3}$

Step 3.2.3

Move $2 x^{3}$ .

$6 h x^{2} + 6 h^{2} x + 2 h^{3} + 2 x^{3}$

Step 3.2.4

Move $6 h x^{2}$ .

$6 h^{2} x + 2 h^{3} + 6 h x^{2} + 2 x^{3}$

Step 3.2.5

Reorder $6 h^{2} x$ and $2 h^{3}$ .

$2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3}$

Step 3.3

Find the components of the definition.

$f (x + h) = 2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3}$

$f (x) = 2 x^{3}$

$f (x + h) = 2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3}$

$f (x) = 2 x^{3}$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3} - (2 x^{3})}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply $- 1$ by $2$ .

$\frac{2 h^{3} + 6 h^{2} x + 6 h x^{2} + 2 x^{3} - 2 x^{3}}{h}$

Step 5.1.2

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{2 h^{3} + 6 h^{2} x + 6 h x^{2} + 0}{h}$

Step 5.1.3

Add $2 h^{3} + 6 h^{2} x + 6 h x^{2}$ and $0$ .

$\frac{2 h^{3} + 6 h^{2} x + 6 h x^{2}}{h}$

Step 5.1.4

Factor $2 h$ out of $2 h^{3} + 6 h^{2} x + 6 h x^{2}$ .

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Step 5.1.4.1

Factor $2 h$ out of $2 h^{3}$ .

$\frac{2 h \cdot h^{2} + 6 h^{2} x + 6 h x^{2}}{h}$

Step 5.1.4.2

Factor $2 h$ out of $6 h^{2} x$ .

$\frac{2 h \cdot h^{2} + 2 h (3 h x) + 6 h x^{2}}{h}$

Step 5.1.4.3

Factor $2 h$ out of $6 h x^{2}$ .

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

Step 5.1.4.4

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

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

Step 5.1.4.5

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

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

Step 5.2

Simplify terms.

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Step 5.2.1

Cancel the common factor of $h$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

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

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

Step 5.2.2

Apply the distributive property.

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply $3$ by $2$ .

$2 h^{2} + 6 (h x) + 2 (3 x^{2})$

Step 5.3.2

Multiply $3$ by $2$ .

$2 h^{2} + 6 (h x) + 6 x^{2}$

Step 5.4

Move $h$ .

$2 h^{2} + 6 x h + 6 x^{2}$

Step 5.5

Move $2 h^{2}$ .

$6 x h + 6 x^{2} + 2 h^{2}$

Step 5.6

Reorder $6 x h$ and $6 x^{2}$ .

$6 x^{2} + 6 x h + 2 h^{2}$

Step 6