Calculus Examples

$4 x \sqrt[3]{x}$

Step 1

Find the first derivative.

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Step 1.1

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

$\frac{d}{d x} [4 x \cdot x^{\frac{1}{3}}]$

Step 1.2

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

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Step 1.2.1

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

$\frac{d}{d x} [4 (x^{\frac{1}{3}} x)]$

Step 1.2.2

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

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Step 1.2.2.1

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

$\frac{d}{d x} [4 (x^{\frac{1}{3}} x^{1})]$

Step 1.2.2.2

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

$\frac{d}{d x} [4 x^{\frac{1}{3} + 1}]$

Step 1.2.3

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

$\frac{d}{d x} [4 x^{\frac{1}{3} + \frac{3}{3}}]$

Step 1.2.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [4 x^{\frac{1 + 3}{3}}]$

Step 1.2.5

Add $1$ and $3$ .

$\frac{d}{d x} [4 x^{\frac{4}{3}}]$

Step 1.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{4}{3}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{4}{3}}]$ .

$4 \frac{d}{d x} [x^{\frac{4}{3}}]$

Step 1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

$4 (\frac{4}{3} x^{\frac{4}{3} - 1})$

Step 1.5

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

$4 (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}})$

Step 1.6

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

$4 (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.7

Combine the numerators over the common denominator.

$4 (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}})$

Step 1.8

Simplify the numerator.

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Step 1.8.1

Multiply $- 1$ by $3$ .

$4 (\frac{4}{3} x^{\frac{4 - 3}{3}})$

Step 1.8.2

Subtract $3$ from $4$ .

$4 (\frac{4}{3} x^{\frac{1}{3}})$

Step 1.9

Combine $\frac{4}{3}$ and $x^{\frac{1}{3}}$ .

$4 \frac{4 x^{\frac{1}{3}}}{3}$

Step 1.10

Combine $4$ and $\frac{4 x^{\frac{1}{3}}}{3}$ .

$\frac{4 (4 x^{\frac{1}{3}})}{3}$

Step 1.11

Multiply $4$ by $4$ .

$f' (x) = \frac{16 x^{\frac{1}{3}}}{3}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{16}{3}$ is constant with respect to $x$ , the derivative of $\frac{16 x^{\frac{1}{3}}}{3}$ with respect to $x$ is $\frac{16}{3} \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$\frac{16}{3} \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 2.2

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

$\frac{16}{3} (\frac{1}{3} x^{\frac{1}{3} - 1})$

Step 2.3

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

$\frac{16}{3} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 2.4

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

$\frac{16}{3} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 2.5

Combine the numerators over the common denominator.

$\frac{16}{3} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

$\frac{16}{3} (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Step 2.6.2

Subtract $3$ from $1$ .

$\frac{16}{3} (\frac{1}{3} x^{\frac{- 2}{3}})$

Step 2.7

Move the negative in front of the fraction.

$\frac{16}{3} (\frac{1}{3} x^{- \frac{2}{3}})$

Step 2.8

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

$\frac{16}{3} \cdot \frac{x^{- \frac{2}{3}}}{3}$

Step 2.9

Multiply $\frac{16}{3}$ by $\frac{x^{- \frac{2}{3}}}{3}$ .

$\frac{16 x^{- \frac{2}{3}}}{3 \cdot 3}$

Step 2.10

Multiply.

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Step 2.10.1

Multiply $3$ by $3$ .

$\frac{16 x^{- \frac{2}{3}}}{9}$

Step 2.10.2

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

$f'' (x) = \frac{16}{9 x^{\frac{2}{3}}}$