Calculus Examples

$\frac{1}{2 \sqrt[3]{x^{5}}}$

Step 1

Rewrite $\frac{d}{d x} [\frac{1}{2 \sqrt[3]{x^{5}}}]$ as $\frac{d}{d x} [\frac{1}{2 (x \sqrt[3]{x^{2}})}]$ .

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Step 1.1

Factor out $x^{3}$ .

$\frac{d}{d x} [\frac{1}{2 \sqrt[3]{x^{3} x^{2}}}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [\frac{1}{2 (x \sqrt[3]{x^{2}})}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\frac{d}{d x} [\frac{1}{2 (x \cdot x^{\frac{2}{3}})}]$

Step 3

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

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

$\frac{d}{d x} [\frac{1}{2 x^{1 + \frac{2}{3}}}]$

Step 3.2

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

$\frac{d}{d x} [\frac{1}{2 x^{\frac{3}{3} + \frac{2}{3}}}]$

Step 3.3

Combine the numerators over the common denominator.

$\frac{d}{d x} [\frac{1}{2 x^{\frac{3 + 2}{3}}}]$

Step 3.4

Add $3$ and $2$ .

$\frac{d}{d x} [\frac{1}{2 x^{\frac{5}{3}}}]$

Step 4

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2 x^{\frac{5}{3}}}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{\frac{5}{3}}}]$ .

$\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{\frac{5}{3}}}]$

Step 5

Apply basic rules of exponents.

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Step 5.1

Rewrite $\frac{1}{x^{\frac{5}{3}}}$ as ${(x^{\frac{5}{3}})}^{- 1}$ .

$\frac{1}{2} \frac{d}{d x} [{(x^{\frac{5}{3}})}^{- 1}]$

Step 5.2

Multiply the exponents in ${(x^{\frac{5}{3}})}^{- 1}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} \frac{d}{d x} [x^{\frac{5}{3} \cdot - 1}]$

Step 5.2.2

Multiply $\frac{5}{3} \cdot - 1$ .

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Step 5.2.2.1

Combine $\frac{5}{3}$ and $- 1$ .

$\frac{1}{2} \frac{d}{d x} [x^{\frac{5 \cdot - 1}{3}}]$

Step 5.2.2.2

Multiply $5$ by $- 1$ .

$\frac{1}{2} \frac{d}{d x} [x^{\frac{- 5}{3}}]$

Step 5.2.3

Move the negative in front of the fraction.

$\frac{1}{2} \frac{d}{d x} [x^{- \frac{5}{3}}]$

Step 6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{5}{3}$ .

$\frac{1}{2} (- \frac{5}{3} x^{- \frac{5}{3} - 1})$

Step 7

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

$\frac{1}{2} (- \frac{5}{3} x^{- \frac{5}{3} - 1 \cdot \frac{3}{3}})$

Step 8

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

$\frac{1}{2} (- \frac{5}{3} x^{- \frac{5}{3} + \frac{- 1 \cdot 3}{3}})$

Step 9

Combine the numerators over the common denominator.

$\frac{1}{2} (- \frac{5}{3} x^{\frac{- 5 - 1 \cdot 3}{3}})$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $3$ .

$\frac{1}{2} (- \frac{5}{3} x^{\frac{- 5 - 3}{3}})$

Step 10.2

Subtract $3$ from $- 5$ .

$\frac{1}{2} (- \frac{5}{3} x^{\frac{- 8}{3}})$

Step 11

Move the negative in front of the fraction.

$\frac{1}{2} (- \frac{5}{3} x^{- \frac{8}{3}})$

Step 12

Combine $x^{- \frac{8}{3}}$ and $\frac{5}{3}$ .

$\frac{1}{2} (- \frac{x^{- \frac{8}{3}} \cdot 5}{3})$

Step 13

Multiply $\frac{1}{2}$ by $\frac{x^{- \frac{8}{3}} \cdot 5}{3}$ .

$- \frac{x^{- \frac{8}{3}} \cdot 5}{2 \cdot 3}$

Step 14

Simplify the expression.

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Step 14.1

Multiply $2$ by $3$ .

$- \frac{x^{- \frac{8}{3}} \cdot 5}{6}$

Step 14.2

Move $5$ to the left of $x^{- \frac{8}{3}}$ .

$- \frac{5 \cdot x^{- \frac{8}{3}}}{6}$

Step 14.3

Move $x^{- \frac{8}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{5}{6 x^{\frac{8}{3}}}$