Calculus Examples

$\frac{9 {(x + h)}^{3} - (9 x^{3})}{h}$

Step 1

Simplify the numerator.

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Step 1.1

Use the Binomial Theorem.

$\frac{9 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) - 1 \cdot 9 x^{3}}{h}$

Step 1.2

Apply the distributive property.

$\frac{9 x^{3} + 9 (3 x^{2} h) + 9 (3 x h^{2}) + 9 h^{3} - 1 \cdot 9 x^{3}}{h}$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $3$ by $9$ .

$\frac{9 x^{3} + 27 (x^{2} h) + 9 (3 x h^{2}) + 9 h^{3} - 1 \cdot 9 x^{3}}{h}$

Step 1.3.2

Multiply $3$ by $9$ .

$\frac{9 x^{3} + 27 (x^{2} h) + 27 (x h^{2}) + 9 h^{3} - 1 \cdot 9 x^{3}}{h}$

Step 1.4

Remove parentheses.

$\frac{9 x^{3} + 27 x^{2} h + 27 x h^{2} + 9 h^{3} - 1 \cdot 9 x^{3}}{h}$

Step 1.5

Multiply $- 1$ by $9$ .

$\frac{9 x^{3} + 27 x^{2} h + 27 x h^{2} + 9 h^{3} - 9 x^{3}}{h}$

Step 1.6

Subtract $9 x^{3}$ from $9 x^{3}$ .

$\frac{27 x^{2} h + 27 x h^{2} + 9 h^{3} + 0}{h}$

Step 1.7

Add $27 x^{2} h + 27 x h^{2} + 9 h^{3}$ and $0$ .

$\frac{27 x^{2} h + 27 x h^{2} + 9 h^{3}}{h}$

Step 1.8

Factor $9 h$ out of $27 x^{2} h + 27 x h^{2} + 9 h^{3}$ .

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Step 1.8.1

Factor $9 h$ out of $27 x^{2} h$ .

$\frac{9 h (3 x^{2}) + 27 x h^{2} + 9 h^{3}}{h}$

Step 1.8.2

Factor $9 h$ out of $27 x h^{2}$ .

$\frac{9 h (3 x^{2}) + 9 h (3 x h) + 9 h^{3}}{h}$

Step 1.8.3

Factor $9 h$ out of $9 h^{3}$ .

$\frac{9 h (3 x^{2}) + 9 h (3 x h) + 9 h (h^{2})}{h}$

Step 1.8.4

Factor $9 h$ out of $9 h (3 x^{2}) + 9 h (3 x h)$ .

$\frac{9 h (3 x^{2} + 3 x h) + 9 h (h^{2})}{h}$

Step 1.8.5

Factor $9 h$ out of $9 h (3 x^{2} + 3 x h) + 9 h (h^{2})$ .

$\frac{9 h (3 x^{2} + 3 x h + h^{2})}{h}$

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.

$\frac{9 h (3 x^{2} + 3 x h + h^{2})}{h}$

Step 2.1.2

Divide $9 (3 x^{2} + 3 x h + h^{2})$ by $1$ .

$9 (3 x^{2} + 3 x h + h^{2})$

Step 2.2

Apply the distributive property.

$9 (3 x^{2}) + 9 (3 x h) + 9 h^{2}$

Step 3

Simplify.

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Step 3.1

Multiply $3$ by $9$ .

$27 x^{2} + 9 (3 x h) + 9 h^{2}$

Step 3.2

Multiply $3$ by $9$ .

$27 x^{2} + 27 (x h) + 9 h^{2}$

Step 4

Move $x$ .

$27 x^{2} + 27 h x + 9 h^{2}$

Step 5

Move $27 x^{2}$ .

$27 h x + 9 h^{2} + 27 x^{2}$

Step 6

Reorder $27 h x$ and $9 h^{2}$ .

$9 h^{2} + 27 h x + 27 x^{2}$