Algebra Examples

$\log_{2} (\sqrt[3]{x}) (x + 1)$

Step 1

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

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

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 $f (x) = \log_{2} (x^{\frac{1}{3}})$ and $g (x) = x + 1$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 3.2

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

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

Step 3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\log_{2} (x^{\frac{1}{3}}) (1 + 0) + (x + 1) \frac{d}{d x} [\log_{2} (x^{\frac{1}{3}})]$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $\log_{2} (x^{\frac{1}{3}})$ by $1$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{2} (x)$ and $g (x) = x^{\frac{1}{3}}$ .

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Step 4.1

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

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

Step 4.2

The derivative of $\log_{2} (u)$ with respect to $u$ is $\frac{1}{u \ln (2)}$ .

$\log_{2} (x^{\frac{1}{3}}) + (x + 1) (\frac{1}{u \ln (2)} \frac{d}{d x} [x^{\frac{1}{3}}])$

Step 4.3

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $3$ .

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

Step 9.2

Subtract $3$ from $1$ .

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

Step 10

Move the negative in front of the fraction.

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Simplify the denominator.

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Step 14.1

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

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Step 14.1.1

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

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

Step 14.1.2

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

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

Step 14.1.3

Combine the numerators over the common denominator.

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

Step 14.1.4

Add $2$ and $1$ .

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

Step 14.1.5

Divide $3$ by $3$ .

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

Step 14.2

Simplify $3 x^{1} \ln (2)$ .

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

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

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

Step 15.2

Combine terms.

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Step 15.2.1

Combine $x$ and $\frac{1}{3 x \ln (2)}$ .

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

Step 15.2.2

Cancel the common factor of $x$ .

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Step 15.2.2.1

Cancel the common factor.

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

Step 15.2.2.2

Rewrite the expression.

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

Step 15.2.3

Multiply $\frac{1}{3 x \ln (2)}$ by $1$ .

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

Step 15.3

Reorder terms.

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