Find the Difference Quotient f(x)=6x^2+4x-3

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

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.

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

Simplify the result.

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

$f (x + h) = 6 (x \cdot x + x h + h (x + h)) + 4 (x + h) - 3$

Apply the distributive property.

$f (x + h) = 6 (x \cdot x + x h + h x + h \cdot h) + 4 (x + h) - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $h$ by $h$ .

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

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

Apply the distributive property.

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

Multiply $2$ by $6$ .

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

Apply the distributive property.

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

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

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

Reorder.

Move $4 x$ .

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

Move $6 x^{2}$ .

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

Reorder $12 h x$ and $6 h^{2}$ .

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

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

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

Simplify.

Multiply $6$ by $- 1$ .

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

Multiply $4$ by $- 1$ .

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

Multiply $- 1$ by $- 3$ .

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

Subtract $6 x^{2}$ from $6 x^{2}$ .

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

Add $6 h^{2}$ and $0$ .

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

Subtract $4 x$ from $4 x$ .

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

Add $6 h^{2}$ and $0$ .

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

Add $- 3$ and $3$ .

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

Add $6 h^{2} + 12 h x + 4 h$ and $0$ .

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

Factor $2 h$ out of $6 h^{2} + 12 h x + 4 h$ .

Factor $2 h$ out of $6 h^{2}$ .

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

Factor $2 h$ out of $12 h x$ .

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

Factor $2 h$ out of $4 h$ .

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

Factor $2 h$ out of $2 h (3 h) + 2 h (6 x)$ .

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

Factor $2 h$ out of $2 h (3 h + 6 x) + 2 h \cdot 2$ .

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

Simplify terms.

Cancel the common factor of $h$ .

Cancel the common factor.

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

Divide $2 (3 h + 6 x + 2)$ by $1$ .

$2 (3 h + 6 x + 2)$

Apply the distributive property.

$2 (3 h) + 2 (6 x) + 2 \cdot 2$

Simplify.

Multiply $3$ by $2$ .

$6 h + 2 (6 x) + 2 \cdot 2$

Multiply $6$ by $2$ .

$6 h + 12 x + 2 \cdot 2$

Multiply $2$ by $2$ .

$6 h + 12 x + 4$

Reorder $6 h$ and $12 x$ .

$12 x + 6 h + 4$

Step 5

Find the Difference Quotient f(x)=6x^2+4x-3

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