Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Combine $\frac{1}{{(1 + x)}^{\frac{1}{2}}}$ and $x$ .

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

Step 1.3

To write $2 {(1 + x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(1 + x)}^{\frac{1}{2}}}{{(1 + x)}^{\frac{1}{2}}}$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Reorder terms.

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

Step 1.6

Rewrite $\frac{2 {(1 + x)}^{\frac{1}{2}} {(1 + x)}^{\frac{1}{2}} - x}{{(x + 1)}^{\frac{1}{2}}}$ in a factored form.

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Step 1.6.1

Multiply ${(1 + x)}^{\frac{1}{2}}$ by ${(1 + x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.6.1.1

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

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

Step 1.6.1.2

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

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

Step 1.6.1.3

Combine the numerators over the common denominator.

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

Step 1.6.1.4

Add $1$ and $1$ .

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

Step 1.6.1.5

Divide $2$ by $2$ .

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

Step 1.6.2

Simplify $2 {(1 + x)}^{1}$ .

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

Step 1.6.3

Apply the distributive property.

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

Step 1.6.4

Multiply $2$ by $1$ .

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

Step 1.6.5

Subtract $x$ from $2 x$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply ${(x + 1)}^{\frac{1}{2}}$ by $x + 1$ by adding the exponents.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.2.4

Add $1$ and $2$ .

$\frac{2 + x}{{(x + 1)}^{\frac{3}{2}}}$