Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Combine $- 1$ and $\frac{\sqrt{1 + x}}{\sqrt{1 + x}}$ .

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 1.4

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

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

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

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

Step 1.5.2

Raise $\sqrt{1 + x}$ to the power of $1$ .

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

Step 1.5.3

Raise $\sqrt{1 + x}$ to the power of $1$ .

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

Step 1.5.4

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

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

Step 1.5.5

Add $1$ and $1$ .

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

Step 1.5.6

Rewrite ${\sqrt{1 + x}}^{2}$ as $1 + x$ .

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Step 1.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + x}$ as ${(1 + x)}^{\frac{1}{2}}$ .

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

Step 1.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.5.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

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

Step 1.5.6.4.2

Rewrite the expression.

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

Step 1.5.6.5

Simplify.

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

Step 1.6

Apply the distributive property.

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

Step 1.7

Multiply $\sqrt{1 + x}$ by $1$ .

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

Step 1.8

Multiply $- \sqrt{1 + x} \sqrt{1 + x}$ .

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Step 1.8.1

Raise $\sqrt{1 + x}$ to the power of $1$ .

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

Step 1.8.2

Raise $\sqrt{1 + x}$ to the power of $1$ .

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

Step 1.8.3

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

$\frac{\frac{\sqrt{1 + x} - {\sqrt{1 + x}}^{1 + 1}}{1 + x}}{x}$

Step 1.8.4

Add $1$ and $1$ .

$\frac{\frac{\sqrt{1 + x} - {\sqrt{1 + x}}^{2}}{1 + x}}{x}$

Step 1.9

Simplify each term.

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Step 1.9.1

Rewrite ${\sqrt{1 + x}}^{2}$ as $1 + x$ .

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Step 1.9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + x}$ as ${(1 + x)}^{\frac{1}{2}}$ .

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

Step 1.9.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.9.1.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.9.1.4

Cancel the common factor of $2$ .

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Step 1.9.1.4.1

Cancel the common factor.

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

Step 1.9.1.4.2

Rewrite the expression.

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

Step 1.9.1.5

Simplify.

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

Step 1.9.2

Apply the distributive property.

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

Step 1.9.3

Multiply $- 1$ by $1$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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

Step 4

Reorder factors in $\frac{\sqrt{1 + x} - 1 - x}{(1 + x) x}$ .

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