Calculus Examples

$\ln (\sqrt{5 x})$

Simplify with factoring out.

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

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

Factor $5$ out of $5 x$ .

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

Apply the product rule to $5 (x)$ .

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

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 $g (x) = 5^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

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

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

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

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

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

Differentiate.

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Since $5^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $5^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $5^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Simplify terms.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

Simplify the expression.

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

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

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

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

Simplify the denominator.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Simplify $2 x^{1}$ .

$\frac{1}{2 x}$