Calculus Examples

$\sqrt[3]{3 x}$

Step 1

Simplify with factoring out.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 x}$ as ${(3 x)}^{\frac{1}{3}}$ .

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

Step 1.2

Factor $3$ out of $3 x$ .

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

Step 1.3

Apply the product rule to $3 (x)$ .

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

Step 2

Since $3^{\frac{1}{3}}$ is constant with respect to $x$ , the derivative of $3^{\frac{1}{3}} x^{\frac{1}{3}}$ with respect to $x$ is $3^{\frac{1}{3}} \frac{d}{d x} [x^{\frac{1}{3}}]$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

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

Step 7.2

Subtract $3$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify the expression.

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Step 11.1

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

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

Step 11.2

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

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

Raise $3$ to the power of $1$ .

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

Step 12.1.2

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

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

Step 12.2

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

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

Step 12.3

Combine the numerators over the common denominator.

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

Step 12.4

Subtract $1$ from $3$ .

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