Calculus Examples

$f (x) = \ln (\frac{\sqrt{x}}{5 - x})$

Step 1

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

$\frac{d}{d x} [\ln (\frac{x^{\frac{1}{2}}}{5 - x})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{x^{\frac{1}{2}}}{5 - x}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{x^{\frac{1}{2}}}{5 - x}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{x^{\frac{1}{2}}}{5 - x}]$

Step 2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [\frac{x^{\frac{1}{2}}}{5 - x}]$

Step 2.3

Replace all occurrences of $u$ with $\frac{x^{\frac{1}{2}}}{5 - x}$ .

$\frac{1}{\frac{x^{\frac{1}{2}}}{5 - x}} \frac{d}{d x} [\frac{x^{\frac{1}{2}}}{5 - x}]$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{x^{\frac{1}{2}}}{5 - x}$ .

$1 \frac{5 - x}{x^{\frac{1}{2}}} \frac{d}{d x} [\frac{x^{\frac{1}{2}}}{5 - x}]$

Step 4

Multiply $\frac{5 - x}{x^{\frac{1}{2}}}$ by $1$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \frac{d}{d x} [\frac{x^{\frac{1}{2}}}{5 - x}]$

Step 5

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 5 - x$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) \frac{d}{d x} [x^{\frac{1}{2}}] - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) (\frac{1}{2} x^{\frac{1}{2} - 1}) - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 8

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

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 9

Combine the numerators over the common denominator.

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) (\frac{1}{2} x^{\frac{1 - 2}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 10.2

Subtract $2$ from $1$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) (\frac{1}{2} x^{\frac{- 1}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) (\frac{1}{2} x^{- \frac{1}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 11.2

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) \frac{x^{- \frac{1}{2}}}{2} - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 11.3

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} \frac{d}{d x} [5 - x]}{{(5 - x)}^{2}}$

Step 12

By the Sum Rule, the derivative of $5 - x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- x]$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} (\frac{d}{d x} [5] + \frac{d}{d x} [- x])}{{(5 - x)}^{2}}$

Step 13

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} (0 + \frac{d}{d x} [- x])}{{(5 - x)}^{2}}$

Step 14

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} \frac{d}{d x} [- x]}{{(5 - x)}^{2}}$

Step 15

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} (- \frac{d}{d x} [x])}{{(5 - x)}^{2}}$

Step 16

Multiply.

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Step 16.1

Multiply $- 1$ by $- 1$ .

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) \frac{1}{2 x^{\frac{1}{2}}} + 1 x^{\frac{1}{2}} \frac{d}{d x} [x]}{{(5 - x)}^{2}}$

Step 16.2

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

$\frac{5 - x}{x^{\frac{1}{2}}} \cdot \frac{(5 - x) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \frac{d}{d x} [x]}{{(5 - x)}^{2}}$

Step 17

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 18

Combine fractions.

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Step 18.1

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

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

Step 18.2

Multiply $\frac{5 - x}{x^{\frac{1}{2}}}$ by $\frac{(5 - x) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}}}{{(5 - x)}^{2}}$ .

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

Step 19

Cancel the common factors.

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Step 19.1

Factor $5 - x$ out of $x^{\frac{1}{2}} {(5 - x)}^{2}$ .

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

Step 19.2

Cancel the common factor.

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

Step 19.3

Rewrite the expression.

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

Step 20

Simplify.

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Step 20.1

Apply the distributive property.

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

Step 20.2

Apply the distributive property.

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

Step 20.3

Simplify the numerator.

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Step 20.3.1

Simplify each term.

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Step 20.3.1.1

Combine $5$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

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

Step 20.3.1.2

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

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

Step 20.3.1.3

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 20.3.1.4

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 20.3.1.4.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 20.3.1.4.1.1

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

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

Step 20.3.1.4.1.2

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

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

Step 20.3.1.4.2

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

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

Step 20.3.1.4.3

Combine the numerators over the common denominator.

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

Step 20.3.1.4.4

Subtract $1$ from $2$ .

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

Step 20.3.2

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

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

Step 20.3.3

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

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

Step 20.3.4

Combine the numerators over the common denominator.

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

Step 20.3.5

Add $- x^{\frac{1}{2}}$ and $x^{\frac{1}{2}} \cdot 2$ .

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Step 20.3.5.1

Reorder $x^{\frac{1}{2}}$ and $2$ .

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

Step 20.3.5.2

Add $- x^{\frac{1}{2}}$ and $2 \cdot x^{\frac{1}{2}}$ .

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

Step 20.4

Combine terms.

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Step 20.4.1

Move $5$ to the left of $x^{\frac{1}{2}}$ .

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

Step 20.4.2

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

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Step 20.4.2.1

Move $x$ .

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

Step 20.4.2.2

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

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Step 20.4.2.2.1

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

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

Step 20.4.2.2.2

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

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

Step 20.4.2.3

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

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

Step 20.4.2.4

Combine the numerators over the common denominator.

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

Step 20.4.2.5

Add $2$ and $1$ .

$\frac{\frac{5}{2 x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}}}{2}}{5 x^{\frac{1}{2}} + x^{\frac{3}{2}} \cdot - 1}$

Step 20.4.3

Move $- 1$ to the left of $x^{\frac{3}{2}}$ .

$\frac{\frac{5}{2 x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}}}{2}}{5 x^{\frac{1}{2}} - 1 \cdot x^{\frac{3}{2}}}$

Step 20.4.4

Rewrite $- 1 x^{\frac{3}{2}}$ as $- x^{\frac{3}{2}}$ .

$\frac{\frac{5}{2 x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}}}{2}}{5 x^{\frac{1}{2}} - x^{\frac{3}{2}}}$

Step 20.4.5

Multiply $\frac{\frac{5}{2 x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}}}{2}}{5 x^{\frac{1}{2}} - x^{\frac{3}{2}}}$ by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} \cdot \frac{\frac{5}{2 x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}}}{2}}{5 x^{\frac{1}{2}} - x^{\frac{3}{2}}}$

Step 20.4.6

Combine.

$\frac{2 x^{\frac{1}{2}} (\frac{5}{2 x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}}}{2})}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.7

Apply the distributive property.

$\frac{2 x^{\frac{1}{2}} \frac{5}{2 x^{\frac{1}{2}}} + 2 x^{\frac{1}{2}} \frac{x^{\frac{1}{2}}}{2}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.8

Cancel the common factor of $2 x^{\frac{1}{2}}$ .

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Step 20.4.8.1

Cancel the common factor.

$\frac{2 x^{\frac{1}{2}} \frac{5}{2 x^{\frac{1}{2}}} + 2 x^{\frac{1}{2}} \frac{x^{\frac{1}{2}}}{2}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.8.2

Rewrite the expression.

$\frac{5 + 2 x^{\frac{1}{2}} \frac{x^{\frac{1}{2}}}{2}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.9

Cancel the common factor of $2$ .

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Step 20.4.9.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{5 + 2 (x^{\frac{1}{2}}) \frac{x^{\frac{1}{2}}}{2}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.9.2

Cancel the common factor.

$\frac{5 + 2 x^{\frac{1}{2}} \frac{x^{\frac{1}{2}}}{2}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.9.3

Rewrite the expression.

$\frac{5 + x^{\frac{1}{2}} x^{\frac{1}{2}}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.10

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

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Step 20.4.10.1

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

$\frac{5 + x^{\frac{1}{2} + \frac{1}{2}}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.10.2

Combine the numerators over the common denominator.

$\frac{5 + x^{\frac{1 + 1}{2}}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.10.3

Add $1$ and $1$ .

$\frac{5 + x^{\frac{2}{2}}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.10.4

Divide $2$ by $2$ .

$\frac{5 + x^{1}}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.4.11

Simplify $x^{1}$ .

$\frac{5 + x}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.5

Reorder terms.

$\frac{x + 5}{2 x^{\frac{1}{2}} (5 x^{\frac{1}{2}} - x^{\frac{3}{2}})}$

Step 20.6

Simplify the denominator.

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Step 20.6.1

Factor $x^{\frac{1}{2}}$ out of $5 x^{\frac{1}{2}} - x^{\frac{3}{2}}$ .

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Step 20.6.1.1

Factor $x^{\frac{1}{2}}$ out of $5 x^{\frac{1}{2}}$ .

$\frac{x + 5}{2 x^{\frac{1}{2}} (x^{\frac{1}{2}} \cdot 5 - x^{\frac{3}{2}})}$

Step 20.6.1.2

Factor $x^{\frac{1}{2}}$ out of $- x^{\frac{3}{2}}$ .

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

Step 20.6.1.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 5 + x^{\frac{1}{2}} (- x^{\frac{2}{2}})$ .

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

Step 20.6.2

Divide $2$ by $2$ .

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

Step 20.6.3

Simplify.

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

Step 20.6.4

Combine exponents.

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Step 20.6.4.1

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

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Step 20.6.4.1.1

Move $x^{\frac{1}{2}}$ .

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

Step 20.6.4.1.2

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

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

Step 20.6.4.1.3

Combine the numerators over the common denominator.

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

Step 20.6.4.1.4

Add $1$ and $1$ .

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

Step 20.6.4.1.5

Divide $2$ by $2$ .

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

Step 20.6.4.2

Simplify $2 x^{1}$ .

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