Precalculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the denominator.

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Step 2.1.2.1.1

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

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Step 2.1.2.1.1.1

Factor $x + h$ out of $3 {(x + h)}^{2}$ .

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

Step 2.1.2.1.1.2

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

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

Step 2.1.2.1.1.3

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

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

Step 2.1.2.1.2

Apply the distributive property.

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

Step 2.1.2.2

Cancel the common factor of $x + h$ .

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Step 2.1.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2

Rewrite the expression.

$f (x + h) = \frac{1}{3 x + 3 h - 5}$

Step 2.1.2.3

The final answer is $\frac{1}{3 x + 3 h - 5}$ .

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

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{1}{3 x + 3 h - 5}$

$f (x) = \frac{1}{3 x - 5}$

$f (x + h) = \frac{1}{3 x + 3 h - 5}$

$f (x) = \frac{1}{3 x - 5}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{1}{3 x + 3 h - 5} - (\frac{1}{3 x - 5})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{1}{3 x + 3 h - 5}$ as a fraction with a common denominator, multiply by $\frac{3 x - 5}{3 x - 5}$ .

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

Step 4.1.2

To write $- \frac{1}{3 x - 5}$ as a fraction with a common denominator, multiply by $\frac{3 x + 3 h - 5}{3 x + 3 h - 5}$ .

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

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

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Step 4.1.5.1

Apply the distributive property.

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

Step 4.1.5.2

Simplify.

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Step 4.1.5.2.1

Multiply $3$ by $- 1$ .

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

Step 4.1.5.2.2

Multiply $3$ by $- 1$ .

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

Step 4.1.5.2.3

Multiply $- 1$ by $- 5$ .

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

Step 4.1.5.3

Subtract $3 x$ from $3 x$ .

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

Step 4.1.5.4

Subtract $5$ from $0$ .

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

Step 4.1.5.5

Add $- 5$ and $5$ .

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

Step 4.1.5.6

Add $- 3 h$ and $0$ .

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

Step 4.1.6

Move the negative in front of the fraction.

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

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

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

Step 4.3.2

Factor $h$ out of $- 3 h$ .

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5