Find the Difference Quotient f(x)=3x^2-x-6

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

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.

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

Simplify the result.

Simplify each term.

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = 3 ((x + h) (x + h)) - (x + h) - 6$

Expand $(x + h) (x + h)$ using the FOIL Method.

Apply the distributive property.

$f (x + h) = 3 (x (x + h) + h (x + h)) - (x + h) - 6$

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h (x + h)) - (x + h) - 6$

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h x + h \cdot h) - (x + h) - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $h$ by $h$ .

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

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

Apply the distributive property.

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

Multiply $2$ by $3$ .

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

Apply the distributive property.

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

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

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

Reorder.

Move $- x$ .

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

Move $3 x^{2}$ .

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

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

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

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

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

Simplify.

Multiply $3$ by $- 1$ .

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

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$\frac{3 h^{2} + 6 h x + 3 x^{2} - h - x - 6 - 3 x^{2} + 1 x - - 6}{h}$

Multiply $x$ by $1$ .

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

Multiply $- 1$ by $- 6$ .

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

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

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

Add $3 h^{2}$ and $0$ .

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

Add $- x$ and $x$ .

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

Add $3 h^{2}$ and $0$ .

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

Add $- 6$ and $6$ .

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

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

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

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

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

$\frac{h (3 h) + 6 h x - h}{h}$

Factor $h$ out of $6 h x$ .

$\frac{h (3 h) + h (6 x) - h}{h}$

Factor $h$ out of $- h$ .

$\frac{h (3 h) + h (6 x) + h \cdot - 1}{h}$

Factor $h$ out of $h (3 h) + h (6 x)$ .

$\frac{h (3 h + 6 x) + h \cdot - 1}{h}$

Factor $h$ out of $h (3 h + 6 x) + h \cdot - 1$ .

$\frac{h (3 h + 6 x - 1)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (3 h + 6 x - 1)}{h}$

Divide $3 h + 6 x - 1$ by $1$ .

$3 h + 6 x - 1$

Reorder $3 h$ and $6 x$ .

$6 x + 3 h - 1$

Step 5

Find the Difference Quotient f(x)=3x^2-x-6

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