Precalculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

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

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Step 2.1.2.1.1

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

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

Step 2.1.2.1.2

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

$f (x + h) = \frac{- {(x + h)}^{2}}{{(x + h)}^{2} (x + h) + {(x + h)}^{2} \cdot - 1}$

Step 2.1.2.1.3

Factor ${(x + h)}^{2}$ out of ${(x + h)}^{2} (x + h) + {(x + h)}^{2} \cdot - 1$ .

$f (x + h) = \frac{- {(x + h)}^{2}}{{(x + h)}^{2} (x + h - 1)}$

Step 2.1.2.2

Cancel the common factor of ${(x + h)}^{2}$ .

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Step 2.1.2.2.1

Cancel the common factor.

$f (x + h) = \frac{- {(x + h)}^{2}}{{(x + h)}^{2} (x + h - 1)}$

Step 2.1.2.2.2

Rewrite the expression.

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

Step 2.1.2.3

Move the negative in front of the fraction.

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

Step 2.1.2.4

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

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- - \frac{1}{x - 1}$ .

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Step 4.1.1.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.1.2

Multiply $\frac{1}{x - 1}$ by $1$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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Step 4.1.4.1

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

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

Step 4.1.4.2

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

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

Step 4.1.4.3

Reorder the factors of $(x - 1) (x + h - 1)$ .

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

Step 4.1.5

Combine the numerators over the common denominator.

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

Step 4.1.6

Rewrite $\frac{- (x - 1) + x + h - 1}{(x + h - 1) (x - 1)}$ in a factored form.

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Step 4.1.6.1

Apply the distributive property.

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

Step 4.1.6.2

Multiply $- 1$ by $- 1$ .

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

Step 4.1.6.3

Add $- x$ and $x$ .

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

Step 4.1.6.4

Add $0$ and $1$ .

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

Step 4.1.6.5

Subtract $1$ from $1$ .

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

Step 4.1.6.6

Add $h$ and $0$ .

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 5