Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $3$ .

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

Step 5.2

Subtract $3$ from $2$ .

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

Step 6

Combine fractions.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Multiply $- 2$ by $1$ .

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

Step 12

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

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

Step 13

Add $2 x - 2$ and $0$ .

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

Step 14

Simplify.

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Step 14.1

Reorder the factors of $\frac{2}{3 {(x^{2} - 2 x - 1)}^{\frac{1}{3}}} (2 x - 2)$ .

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

Step 14.2

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

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

Step 14.3

Simplify the numerator.

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Step 14.3.1

Factor $2$ out of $2 x - 2$ .

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Step 14.3.1.1

Factor $2$ out of $2 x$ .

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

Step 14.3.1.2

Factor $2$ out of $- 2$ .

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

Step 14.3.1.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

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

Step 14.3.2

Multiply $2$ by $2$ .

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