Find the Difference Quotient f(x)=4x^3-6

$f (x) = 4 x^{3} - 6$

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.

$f (x + h) = 4 {(x + h)}^{3} - 6$

Simplify the result.

Simplify each term.

Use the Binomial Theorem.

$f (x + h) = 4 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) - 6$

Apply the distributive property.

$f (x + h) = 4 x^{3} + 4 (3 x^{2} h) + 4 (3 x h^{2}) + 4 h^{3} - 6$

Simplify.

Multiply $3$ by $4$ .

$f (x + h) = 4 x^{3} + 12 (x^{2} h) + 4 (3 x h^{2}) + 4 h^{3} - 6$

Multiply $3$ by $4$ .

$f (x + h) = 4 x^{3} + 12 (x^{2} h) + 12 (x h^{2}) + 4 h^{3} - 6$

Remove parentheses.

$f (x + h) = 4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} - 6$

The final answer is $4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} - 6$ .

$4 x^{3} + 12 x^{2} h + 12 x h^{2} + 4 h^{3} - 6$

Reorder.

Move $x^{2}$ .

$4 x^{3} + 12 h x^{2} + 12 x h^{2} + 4 h^{3} - 6$

Move $x$ .

$4 x^{3} + 12 h x^{2} + 12 h^{2} x + 4 h^{3} - 6$

Move $4 x^{3}$ .

$12 h x^{2} + 12 h^{2} x + 4 h^{3} + 4 x^{3} - 6$

Move $12 h x^{2}$ .

$12 h^{2} x + 4 h^{3} + 12 h x^{2} + 4 x^{3} - 6$

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

$4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} - 6$

Find the components of the definition.

$f (x + h) = 4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} - 6$

$f (x) = 4 x^{3} - 6$

$f (x + h) = 4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} - 6$

$f (x) = 4 x^{3} - 6$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} - 6 - (4 x^{3} - 6)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} - 6 - (4 x^{3}) - - 6}{h}$

Multiply $4$ by $- 1$ .

$\frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} - 6 - 4 x^{3} - - 6}{h}$

Multiply $- 1$ by $- 6$ .

$\frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 4 x^{3} - 6 - 4 x^{3} + 6}{h}$

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

$\frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} + 0 - 6 + 6}{h}$

Add $4 h^{3}$ and $0$ .

$\frac{4 h^{3} + 12 h^{2} x + 12 h x^{2} - 6 + 6}{h}$

Add $- 6$ and $6$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify terms.

Cancel the common factor of $h$ .

Cancel the common factor.

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $3$ by $4$ .

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

Multiply $3$ by $4$ .

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

Move $h$ .

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

Move $4 h^{2}$ .

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

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

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

Step 5

Find the Difference Quotient f(x)=4x^3-6

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