Calculus Examples

$\sqrt[3]{x} (x + 5)$

Step 1

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

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = x + 5$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

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

Step 3.2

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

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

Step 3.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$x^{\frac{1}{3}} (1 + 0) + (x + 5) \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

$x^{\frac{1}{3}} \cdot 1 + (x + 5) \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 3.4.2

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

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

Step 3.5

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

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

Step 7.2

Subtract $3$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$x^{\frac{1}{3}} + x \frac{1}{3 x^{\frac{2}{3}}} + 5 \frac{1}{3 x^{\frac{2}{3}}}$

Step 11.2

Combine terms.

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Step 11.2.1

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

$x^{\frac{1}{3}} + \frac{x}{3 x^{\frac{2}{3}}} + 5 \frac{1}{3 x^{\frac{2}{3}}}$

Step 11.2.2

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

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

Step 11.2.3

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

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Step 11.2.3.1

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

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Step 11.2.3.1.1

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

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

Step 11.2.3.1.2

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

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

Step 11.2.3.2

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

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

Step 11.2.3.3

Combine the numerators over the common denominator.

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

Step 11.2.3.4

Subtract $2$ from $3$ .

$x^{\frac{1}{3}} + \frac{x^{\frac{1}{3}}}{3} + 5 \frac{1}{3 x^{\frac{2}{3}}}$

Step 11.2.4

Combine $5$ and $\frac{1}{3 x^{\frac{2}{3}}}$ .

$x^{\frac{1}{3}} + \frac{x^{\frac{1}{3}}}{3} + \frac{5}{3 x^{\frac{2}{3}}}$

Step 11.2.5

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

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

Step 11.2.6

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

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

Step 11.2.7

Combine the numerators over the common denominator.

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

Step 11.2.8

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

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

Step 11.2.9

Add $3 x^{\frac{1}{3}}$ and $x^{\frac{1}{3}}$ .

$\frac{4 x^{\frac{1}{3}}}{3} + \frac{5}{3 x^{\frac{2}{3}}}$