Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

To write $\frac{5}{{(x + h)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

Multiply $\frac{5}{{(x + h)}^{2}}$ by $\frac{x^{2}}{x^{2}}$ .

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

Step 1.3.2

Multiply $\frac{5}{x^{2}}$ by $\frac{{(x + h)}^{2}}{{(x + h)}^{2}}$ .

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

Step 1.3.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

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

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Step 1.5.1

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

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Step 1.5.1.1

Factor $5$ out of $5 x^{2}$ .

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

Step 1.5.1.2

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

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

Step 1.5.1.3

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

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

Step 1.5.2

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 $b = x + h$ .

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

Add $x$ and $x$ .

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

Step 1.5.3.2

Apply the distributive property.

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

Step 1.5.3.3

Subtract $x$ from $x$ .

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

Step 1.5.3.4

Subtract $h$ from $0$ .

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

Step 1.5.3.5

Factor out negative.

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

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

Step 1.5.4

Combine exponents.

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Step 1.5.4.1

Factor out negative.

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

Step 1.5.4.2

Multiply $5$ by $- 1$ .

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

Step 1.6

Move the negative in front of the fraction.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Cancel the common factor of $h$ .

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Step 3.1

Move the leading negative in $- \frac{(5 (2 x + h)) h}{x^{2} {(x + h)}^{2}}$ into the numerator.

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

Step 3.2

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

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

Step 3.3

Cancel the common factor.

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

Step 3.4

Rewrite the expression.

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

Step 4

Move the negative in front of the fraction.

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