Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite ${(x + h)}^{- 2}$ as ${({(x + h)}^{- 1})}^{2}$ .

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

Step 1.2

Rewrite $x^{- 2}$ as ${(x^{- 1})}^{2}$ .

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

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x + h)}^{- 1}$ and $b = x^{- 1}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

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

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Step 1.4.5.1

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

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

Step 1.4.5.2

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

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

Step 1.4.5.3

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

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

Step 1.4.6

Combine the numerators over the common denominator.

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

Step 1.4.7

Add $x$ and $x$ .

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

Step 1.4.8

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.9

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.10

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

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

Step 1.4.11

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

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

Step 1.4.12

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

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Step 1.4.12.1

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

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

Step 1.4.12.2

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

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

Step 1.4.12.3

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

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

Step 1.4.13

Combine the numerators over the common denominator.

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

Step 1.4.14

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

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Step 1.4.14.1

Apply the distributive property.

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

Step 1.4.14.2

Subtract $x$ from $x$ .

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

Step 1.4.14.3

Subtract $h$ from $0$ .

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

Step 1.5

Move the negative in front of the fraction.

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

Step 1.6

Combine exponents.

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Step 1.6.1

Factor out negative.

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

Step 1.6.2

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

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

Step 1.6.3

Raise $x$ to the power of $1$ .

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

Step 1.6.4

Raise $x$ to the power of $1$ .

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

Step 1.6.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.6.6

Add $1$ and $1$ .

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

Step 1.6.7

Raise $x + h$ to the power of $1$ .

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

Step 1.6.8

Raise $x + h$ to the power of $1$ .

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

Step 1.6.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.6.10

Add $1$ and $1$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Cancel the common factor of $h$ .

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Step 3.1

Move the leading negative in $- \frac{(2 x + h) h}{x^{2} {(x + h)}^{2}}$ into the numerator.

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

Step 3.2

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

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

Step 3.3

Cancel the common factor.

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

Step 3.4

Rewrite the expression.

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

Step 4

Move the negative in front of the fraction.

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