Find the Difference Quotient f(x)=9x^3

$f (x) = 9 x^{3}$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

Evaluate the function at $x = x + h$ .

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 9 {(x + h)}^{3}$

Simplify the result.

Use the Binomial Theorem.

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

Apply the distributive property.

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

Simplify.

Multiply $3$ by $9$ .

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

Multiply $3$ by $9$ .

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

Remove parentheses.

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

The final answer is $9 x^{3} + 27 x^{2} h + 27 x h^{2} + 9 h^{3}$ .

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

Reorder.

Move $x^{2}$ .

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

Move $x$ .

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

Move $9 x^{3}$ .

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

Move $27 h x^{2}$ .

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

Reorder $27 h^{2} x$ and $9 h^{3}$ .

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

Find the components of the definition.

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

$f (x) = 9 x^{3}$

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

$f (x) = 9 x^{3}$

Step 3

Plug in the components.

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

Step 4

Simplify.

Simplify the numerator.

Multiply $- 1$ by $9$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify terms.

Cancel the common factor of $h$ .

Cancel the common factor.

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $3$ by $9$ .

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

Multiply $3$ by $9$ .

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

Move $h$ .

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

Move $9 h^{2}$ .

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

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

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

Step 5

Find the Difference Quotient f(x)=9x^3

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