Precalculus Examples

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

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

The final answer is $- \frac{2}{{(x + h)}^{2}}$ .

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

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

Tap for more steps...

Step 4.1.1.1

Move parentheses.

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

Step 4.1.1.2

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

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

Step 4.1.1.3

Factor $- 1$ out of $- - 1 \frac{2}{x^{2}}$ .

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

Step 4.1.1.4

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

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

Step 4.1.2

To write $\frac{2}{{(x + h)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

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

Step 4.1.3

To write $- \frac{2}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x + h)}^{2}}{{(x + h)}^{2}}$ .

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

Step 4.1.4

Write each expression with a common denominator of ${(x + h)}^{2} x^{2}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1.4.1

Multiply $\frac{2}{{(x + h)}^{2}}$ by $\frac{x^{2}}{x^{2}}$ .

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

Step 4.1.4.2

Multiply $\frac{2}{x^{2}}$ by $\frac{{(x + h)}^{2}}{{(x + h)}^{2}}$ .

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

Step 4.1.4.3

Reorder the factors of ${(x + h)}^{2} x^{2}$ .

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

Step 4.1.5

Combine the numerators over the common denominator.

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

Step 4.1.6

Simplify the numerator.

Tap for more steps...

Step 4.1.6.1

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

Tap for more steps...

Step 4.1.6.1.1

Factor $2$ out of $2 x^{2}$ .

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

Step 4.1.6.1.2

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

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

Step 4.1.6.1.3

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

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

Step 4.1.6.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = x + h$ .

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

Step 4.1.6.3

Simplify.

Tap for more steps...

Step 4.1.6.3.1

Add $x$ and $x$ .

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

Step 4.1.6.3.2

Apply the distributive property.

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

Step 4.1.6.3.3

Subtract $x$ from $x$ .

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

Step 4.1.6.3.4

Subtract $h$ from $0$ .

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

Step 4.1.6.3.5

Factor out negative.

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

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

Step 4.1.6.4

Combine exponents.

Tap for more steps...

Step 4.1.6.4.1

Factor out negative.

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

Step 4.1.6.4.2

Multiply $2$ by $- 1$ .

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

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

Step 4.1.7

Move the negative in front of the fraction.

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

Step 4.1.8

Combine exponents.

Tap for more steps...

Step 4.1.8.1

Factor out negative.

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

Step 4.1.8.2

Multiply $- 1$ by $- 1$ .

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

Step 4.1.8.3

Multiply $\frac{2 (2 x + h) h}{x^{2} {(x + h)}^{2}}$ by $1$ .

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Combine.

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

Step 4.4

Cancel the common factor of $h$ .

Tap for more steps...

Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Multiply $2$ by $1$ .

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

Step 5