Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Use the Binomial Theorem.

$\frac{4 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) - 1 \cdot 4 x^{3}}{h}$

Step 1.2

Apply the distributive property.

$\frac{4 x^{3} + 4 (3 x^{2} h) + 4 (3 x h^{2}) + 4 h^{3} - 1 \cdot 4 x^{3}}{h}$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $3$ by $4$ .

$\frac{4 x^{3} + 12 (x^{2} h) + 4 (3 x h^{2}) + 4 h^{3} - 1 \cdot 4 x^{3}}{h}$

Step 1.3.2

Multiply $3$ by $4$ .

$\frac{4 x^{3} + 12 (x^{2} h) + 12 (x h^{2}) + 4 h^{3} - 1 \cdot 4 x^{3}}{h}$

Step 1.4

Remove parentheses.

$\frac{4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} - 1 \cdot 4 x^{3}}{h}$

Step 1.5

Multiply $- 1$ by $4$ .

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

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

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

Step 1.8.5

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

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

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $h$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

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

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

Step 2.2

Apply the distributive property.

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

Step 3

Simplify.

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Step 3.1

Multiply $3$ by $4$ .

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

Step 3.2

Multiply $3$ by $4$ .

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

Step 4

Move $x$ .

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

Step 5

Move $12 x^{2}$ .

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

Step 6

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

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