Precalculus Examples

$f (x) = \frac{- 2}{3 - 5 x}$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \frac{- 2}{3 - 5 (x + h)}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = \frac{- 2}{3 - 5 x - 5 h}$

Step 2.1.2.2

Move the negative in front of the fraction.

$f (x + h) = - \frac{2}{3 - 5 x - 5 h}$

Step 2.1.2.3

The final answer is $- \frac{2}{3 - 5 x - 5 h}$ .

$- \frac{2}{3 - 5 x - 5 h}$

Step 2.2

Find the components of the definition.

$f (x + h) = - \frac{2}{3 - 5 x - 5 h}$

$f (x) = - \frac{2}{3 - 5 x}$

$f (x + h) = - \frac{2}{3 - 5 x - 5 h}$

$f (x) = - \frac{2}{3 - 5 x}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{2}{3 - 5 x - 5 h} - (- \frac{2}{3 - 5 x})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- - \frac{2}{3 - 5 x}$ .

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Step 4.1.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{2}{3 - 5 x - 5 h} + 1 \frac{2}{3 - 5 x}}{h}$

Step 4.1.1.2

Multiply $\frac{2}{3 - 5 x}$ by $1$ .

$\frac{- \frac{2}{3 - 5 x - 5 h} + \frac{2}{3 - 5 x}}{h}$

Step 4.1.2

To write $- \frac{2}{3 - 5 x - 5 h}$ as a fraction with a common denominator, multiply by $\frac{3 - 5 x}{3 - 5 x}$ .

$\frac{- \frac{2}{3 - 5 x - 5 h} \cdot \frac{3 - 5 x}{3 - 5 x} + \frac{2}{3 - 5 x}}{h}$

Step 4.1.3

To write $\frac{2}{3 - 5 x}$ as a fraction with a common denominator, multiply by $\frac{3 - 5 x - 5 h}{3 - 5 x - 5 h}$ .

$\frac{- \frac{2}{3 - 5 x - 5 h} \cdot \frac{3 - 5 x}{3 - 5 x} + \frac{2}{3 - 5 x} \cdot \frac{3 - 5 x - 5 h}{3 - 5 x - 5 h}}{h}$

Step 4.1.4

Write each expression with a common denominator of $(3 - 5 x - 5 h) (3 - 5 x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.4.1

Multiply $\frac{2}{3 - 5 x - 5 h}$ by $\frac{3 - 5 x}{3 - 5 x}$ .

$\frac{- \frac{2 (3 - 5 x)}{(3 - 5 x - 5 h) (3 - 5 x)} + \frac{2}{3 - 5 x} \cdot \frac{3 - 5 x - 5 h}{3 - 5 x - 5 h}}{h}$

Step 4.1.4.2

Multiply $\frac{2}{3 - 5 x}$ by $\frac{3 - 5 x - 5 h}{3 - 5 x - 5 h}$ .

$\frac{- \frac{2 (3 - 5 x)}{(3 - 5 x - 5 h) (3 - 5 x)} + \frac{2 (3 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.4.3

Reorder the factors of $(3 - 5 x - 5 h) (3 - 5 x)$ .

$\frac{- \frac{2 (3 - 5 x)}{(3 - 5 x) (3 - 5 x - 5 h)} + \frac{2 (3 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{\frac{- 2 (3 - 5 x) + 2 (3 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6

Rewrite $\frac{- 2 (3 - 5 x) + 2 (3 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}$ in a factored form.

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Step 4.1.6.1

Factor $2$ out of $- 2 (3 - 5 x) + 2 (3 - 5 x - 5 h)$ .

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Step 4.1.6.1.1

Factor $2$ out of $- 2 (3 - 5 x)$ .

$\frac{\frac{2 (- (3 - 5 x)) + 2 (3 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6.1.2

Factor $2$ out of $2 (- (3 - 5 x)) + 2 (3 - 5 x - 5 h)$ .

$\frac{\frac{2 (- (3 - 5 x) + 3 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6.2

Apply the distributive property.

$\frac{\frac{2 (- 1 \cdot 3 - (- 5 x) + 3 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6.3

Multiply $- 1$ by $3$ .

$\frac{\frac{2 (- 3 - (- 5 x) + 3 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6.4

Multiply $- 5$ by $- 1$ .

$\frac{\frac{2 (- 3 + 5 x + 3 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6.5

Add $- 3$ and $3$ .

$\frac{\frac{2 (5 x + 0 - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6.6

Add $5 x$ and $0$ .

$\frac{\frac{2 (5 x - 5 x - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6.7

Subtract $5 x$ from $5 x$ .

$\frac{\frac{2 (0 - 5 h)}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6.8

Subtract $5 h$ from $0$ .

$\frac{\frac{2 \cdot - 5 h}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.6.9

Multiply $2$ by $- 5$ .

$\frac{\frac{- 10 h}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.1.7

Move the negative in front of the fraction.

$\frac{- \frac{10 h}{(3 - 5 x) (3 - 5 x - 5 h)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{10 h}{(3 - 5 x) (3 - 5 x - 5 h)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{10 h}{(3 - 5 x) (3 - 5 x - 5 h)}$ into the numerator.

$\frac{- 10 h}{(3 - 5 x) (3 - 5 x - 5 h)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 10 h$ .

$\frac{h \cdot - 10}{(3 - 5 x) (3 - 5 x - 5 h)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 10}{(3 - 5 x) (3 - 5 x - 5 h)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 10}{(3 - 5 x) (3 - 5 x - 5 h)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{10}{(3 - 5 x) (3 - 5 x - 5 h)}$

Step 5