Precalculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{\frac{2}{3}} - 1$ .

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 1.3

Replace all occurrences of $u$ with $x^{\frac{2}{3}} - 1$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{\frac{2}{3}} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- 1]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $2$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

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

$2 (x^{\frac{2}{3}} - 1) (\frac{2}{3 x^{\frac{1}{3}}} + 0)$

Step 9

Combine fractions.

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Step 9.1

Add $\frac{2}{3 x^{\frac{1}{3}}}$ and $0$ .

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

Step 9.2

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

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

Step 9.3

Multiply $2$ by $2$ .

$\frac{4}{3 x^{\frac{1}{3}}} (x^{\frac{2}{3}} - 1)$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Combine terms.

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Step 10.2.1

Combine $\frac{4}{3 x^{\frac{1}{3}}}$ and $x^{\frac{2}{3}}$ .

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

Step 10.2.2

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

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

Step 10.2.3

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

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Step 10.2.3.1

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

$\frac{4 (x^{- \frac{1}{3}} x^{\frac{2}{3}})}{3} + \frac{4}{3 x^{\frac{1}{3}}} \cdot - 1$

Step 10.2.3.2

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

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

Step 10.2.3.3

Combine the numerators over the common denominator.

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

Step 10.2.3.4

Add $- 1$ and $2$ .

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

Step 10.2.4

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

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

Step 10.2.5

Multiply $4$ by $- 1$ .

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

Step 10.2.6

Move the negative in front of the fraction.

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