Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Add $x$ and $x$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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