Find the Difference Quotient f(x)=9x^2-8

$f (x) = 9 x^{2} - 8$

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.

$f (x + h) = 9 {(x + h)}^{2} - 8$

Simplify the result.

Simplify each term.

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

$f (x + h) = 9 ((x + h) (x + h)) - 8$

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

Apply the distributive property.

$f (x + h) = 9 (x (x + h) + h (x + h)) - 8$

Apply the distributive property.

$f (x + h) = 9 (x \cdot x + x h + h (x + h)) - 8$

Apply the distributive property.

$f (x + h) = 9 (x \cdot x + x h + h x + h \cdot h) - 8$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = 9 (x^{2} + x h + h x + h \cdot h) - 8$

Multiply $h$ by $h$ .

$f (x + h) = 9 (x^{2} + x h + h x + h^{2}) - 8$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = 9 (x^{2} + h x + h x + h^{2}) - 8$

Add $h x$ and $h x$ .

$f (x + h) = 9 (x^{2} + 2 h x + h^{2}) - 8$

Apply the distributive property.

$f (x + h) = 9 x^{2} + 9 (2 h x) + 9 h^{2} - 8$

Multiply $2$ by $9$ .

$f (x + h) = 9 x^{2} + 18 h x + 9 h^{2} - 8$

The final answer is $9 x^{2} + 18 h x + 9 h^{2} - 8$ .

$9 x^{2} + 18 h x + 9 h^{2} - 8$

Reorder.

Move $9 x^{2}$ .

$18 h x + 9 h^{2} + 9 x^{2} - 8$

Reorder $18 h x$ and $9 h^{2}$ .

$9 h^{2} + 18 h x + 9 x^{2} - 8$

Find the components of the definition.

$f (x + h) = 9 h^{2} + 18 h x + 9 x^{2} - 8$

$f (x) = 9 x^{2} - 8$

$f (x + h) = 9 h^{2} + 18 h x + 9 x^{2} - 8$

$f (x) = 9 x^{2} - 8$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{9 h^{2} + 18 h x + 9 x^{2} - 8 - (9 x^{2} - 8)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{9 h^{2} + 18 h x + 9 x^{2} - 8 - (9 x^{2}) - - 8}{h}$

Multiply $9$ by $- 1$ .

$\frac{9 h^{2} + 18 h x + 9 x^{2} - 8 - 9 x^{2} - - 8}{h}$

Multiply $- 1$ by $- 8$ .

$\frac{9 h^{2} + 18 h x + 9 x^{2} - 8 - 9 x^{2} + 8}{h}$

Subtract $9 x^{2}$ from $9 x^{2}$ .

$\frac{9 h^{2} + 18 h x + 0 - 8 + 8}{h}$

Add $9 h^{2}$ and $0$ .

$\frac{9 h^{2} + 18 h x - 8 + 8}{h}$

Add $- 8$ and $8$ .

$\frac{9 h^{2} + 18 h x + 0}{h}$

Add $9 h^{2} + 18 h x$ and $0$ .

$\frac{9 h^{2} + 18 h x}{h}$

Factor $9 h$ out of $9 h^{2} + 18 h x$ .

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

$\frac{9 h \cdot h + 18 h x}{h}$

Factor $9 h$ out of $18 h x$ .

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

Factor $9 h$ out of $9 h \cdot h + 9 h (2 x)$ .

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

Simplify terms.

Cancel the common factor of $h$ .

Cancel the common factor.

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

Divide $9 (h + 2 x)$ by $1$ .

$9 (h + 2 x)$

Apply the distributive property.

$9 h + 9 (2 x)$

Simplify the expression.

Multiply $2$ by $9$ .

$9 h + 18 x$

Reorder $9 h$ and $18 x$ .

$18 x + 9 h$

Step 5

Find the Difference Quotient f(x)=9x^2-8

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