Find the Difference Quotient f(x)=-1/(9x)

$f (x) = - \frac{1}{9 x}$

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.

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

The final answer is $- \frac{1}{9 (x + h)}$ .

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

Find the components of the definition.

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

$f (x) = - \frac{1}{9 x}$

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

$f (x) = - \frac{1}{9 x}$

Step 3

Plug in the components.

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

Step 4

Simplify.

Simplify the numerator.

Factor $- 1$ out of $- \frac{1}{9 (x + h)} - - \frac{1}{9 x}$ .

Move parentheses.

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

Factor $- 1$ out of $- \frac{1}{9 (x + h)}$ .

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

Factor $- 1$ out of $- - 1 \frac{1}{9 x}$ .

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

Factor $- 1$ out of $- \frac{1}{9 (x + h)} - - \frac{1}{9 x}$ .

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

To write $\frac{1}{9 (x + h)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

To write $- \frac{1}{9 x}$ as a fraction with a common denominator, multiply by $\frac{x + h}{x + h}$ .

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

Write each expression with a common denominator of $9 (x + h) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{9 (x + h)}$ by $\frac{x}{x}$ .

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

Multiply $\frac{1}{9 x}$ by $\frac{x + h}{x + h}$ .

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

Reorder the factors of $9 (x + h) x$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Subtract $x$ from $x$ .

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

Subtract $h$ from $0$ .

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

Move the negative in front of the fraction.

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

Combine exponents.

Factor out negative.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{h}{9 x (x + h)}$ by $1$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Rewrite the expression.

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

Step 5

Find the Difference Quotient f(x)=-1/(9x)

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