Calculus Examples

$f (x) = \ln (\sqrt{x})$

Step 1

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

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

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify the expression.

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Step 11.1

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

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

Step 11.2

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

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

Step 12

Simplify the denominator.

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Step 12.1

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

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.1.3

Combine the numerators over the common denominator.

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

Step 12.1.4

Add $1$ and $1$ .

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

Step 12.1.5

Divide $2$ by $2$ .

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

Step 12.2

Simplify $2 x^{1}$ .

$\frac{1}{2 x}$