Calculus Examples

$\ln ({(x)}^{\frac{1}{3}})$

Step 1

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 $g (x) = {(x)}^{\frac{1}{3}}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as ${(x)}^{\frac{1}{3}}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [{(x)}^{\frac{1}{3}}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [{(x)}^{\frac{1}{3}}]$

Step 1.3

Replace all occurrences of $u$ with ${(x)}^{\frac{1}{3}}$ .

$\frac{1}{x^{\frac{1}{3}}} \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{3}$ .

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

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Combine $\frac{1}{3}$ and $x^{- \frac{2}{3}}$ .

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

Step 9

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

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

Step 10

Simplify the expression.

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Step 10.1

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

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

Step 10.2

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 x^{\frac{1}{3}} x^{\frac{2}{3}}}$

Step 11

Simplify the denominator.

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Step 11.1

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 11.1.1

Move $x^{\frac{2}{3}}$ .

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

Step 11.1.2

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

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

Step 11.1.3

Combine the numerators over the common denominator.

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

Step 11.1.4

Add $2$ and $1$ .

$\frac{1}{3 x^{\frac{3}{3}}}$

Step 11.1.5

Divide $3$ by $3$ .

$\frac{1}{3 x^{1}}$

Step 11.2

Simplify $3 x^{1}$ .

$\frac{1}{3 x}$