Find the Difference Quotient f(x)=-9x+9

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

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.

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

Simplify the result.

Apply the distributive property.

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

The final answer is $- 9 x - 9 h + 9$ .

$- 9 x - 9 h + 9$

Reorder $- 9 x$ and $- 9 h$ .

$- 9 h - 9 x + 9$

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 9 h - 9 x + 9 - (- 9 x + 9)}{h}$

Step 4

Simplify.

Simplify the numerator.

Factor $9$ out of $- 9 x + 9$ .

Factor $9$ out of $- 9 x$ .

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

Factor $9$ out of $9$ .

$\frac{- 9 h - 9 x + 9 - (9 (- x) + 9 (1))}{h}$

Factor $9$ out of $9 (- x) + 9 (1)$ .

$\frac{- 9 h - 9 x + 9 - (9 (- x + 1))}{h}$

$\frac{- 9 h - 9 x + 9 - 1 \cdot 9 (- x + 1)}{h}$

Multiply $- 1$ by $9$ .

$\frac{- 9 h - 9 x + 9 - 9 (- x + 1)}{h}$

Factor $9$ out of $- 9 h - 9 x + 9 - 9 (- x + 1)$ .

Factor $9$ out of $- 9 h$ .

$\frac{9 (- h) - 9 x + 9 - 9 (- x + 1)}{h}$

Factor $9$ out of $- 9 x$ .

$\frac{9 (- h) + 9 (- x) + 9 - 9 (- x + 1)}{h}$

Factor $9$ out of $9$ .

$\frac{9 (- h) + 9 (- x) + 9 (1) - 9 (- x + 1)}{h}$

Factor $9$ out of $- 9 (- x + 1)$ .

$\frac{9 (- h) + 9 (- x) + 9 (1) + 9 (- (- x + 1))}{h}$

Factor $9$ out of $9 (- h) + 9 (- x)$ .

$\frac{9 (- h - x) + 9 (1) + 9 (- (- x + 1))}{h}$

Factor $9$ out of $9 (- h - x) + 9 (1)$ .

$\frac{9 (- h - x + 1) + 9 (- (- x + 1))}{h}$

Factor $9$ out of $9 (- h - x + 1) + 9 (- (- x + 1))$ .

$\frac{9 (- h - x + 1 - (- x + 1))}{h}$

Apply the distributive property.

$\frac{9 (- h - x + 1 - - x - 1 \cdot 1)}{h}$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$\frac{9 (- h - x + 1 + 1 x - 1 \cdot 1)}{h}$

Multiply $x$ by $1$ .

$\frac{9 (- h - x + 1 + x - 1 \cdot 1)}{h}$

Multiply $- 1$ by $1$ .

$\frac{9 (- h - x + 1 + x - 1)}{h}$

Add $- x$ and $x$ .

$\frac{9 (- h + 0 + 1 - 1)}{h}$

Add $- h$ and $0$ .

$\frac{9 (- h + 1 - 1)}{h}$

Subtract $1$ from $1$ .

$\frac{9 (- h + 0)}{h}$

Add $- h$ and $0$ .

$\frac{9 \cdot - 1 h}{h}$

Combine exponents.

Factor out negative.

$\frac{- (9 h)}{h}$

Multiply $9$ by $- 1$ .

$\frac{- 9 h}{h}$

Divide $- 9$ by $1$ .

$- 9$

Step 5

Find the Difference Quotient f(x)=-9x+9

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