Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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

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

Step 3

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 $g (x) = x^{2} - 2$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $3$ .

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

Step 7.2

Subtract $3$ from $2$ .

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

Step 8

Combine fractions.

Tap for more steps...

Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

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

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

Step 10

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

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

Step 11

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 12

Combine fractions.

Tap for more steps...

Step 12.1

Add $2 x$ and $0$ .

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

Step 12.2

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

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

Step 12.3

Multiply $2$ by $2$ .

$\frac{1}{x {(x^{2} - 2)}^{\frac{2}{3}}} (\frac{4 x}{3 {(x^{2} - 2)}^{\frac{1}{3}}} x + {(x^{2} - 2)}^{\frac{2}{3}} \frac{d}{d x} [x])$

Step 12.4

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

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

Step 13

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

$\frac{1}{x {(x^{2} - 2)}^{\frac{2}{3}}} (\frac{4 (x^{1} x)}{3 {(x^{2} - 2)}^{\frac{1}{3}}} + {(x^{2} - 2)}^{\frac{2}{3}} \frac{d}{d x} [x])$

Step 14

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

$\frac{1}{x {(x^{2} - 2)}^{\frac{2}{3}}} (\frac{4 (x^{1} x^{1})}{3 {(x^{2} - 2)}^{\frac{1}{3}}} + {(x^{2} - 2)}^{\frac{2}{3}} \frac{d}{d x} [x])$

Step 15

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

$\frac{1}{x {(x^{2} - 2)}^{\frac{2}{3}}} (\frac{4 x^{1 + 1}}{3 {(x^{2} - 2)}^{\frac{1}{3}}} + {(x^{2} - 2)}^{\frac{2}{3}} \frac{d}{d x} [x])$

Step 16

Add $1$ and $1$ .

$\frac{1}{x {(x^{2} - 2)}^{\frac{2}{3}}} (\frac{4 x^{2}}{3 {(x^{2} - 2)}^{\frac{1}{3}}} + {(x^{2} - 2)}^{\frac{2}{3}} \frac{d}{d x} [x])$

Step 17

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

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

Step 18

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

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

Step 19

To write ${(x^{2} - 2)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x^{2} - 2)}^{\frac{1}{3}}}{3 {(x^{2} - 2)}^{\frac{1}{3}}}$ .

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

Step 20

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

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

Step 21

Combine the numerators over the common denominator.

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

Step 22

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

Tap for more steps...

Step 22.1

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

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

Step 22.2

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

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

Step 22.3

Combine the numerators over the common denominator.

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

Step 22.4

Add $1$ and $2$ .

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

Step 22.5

Divide $3$ by $3$ .

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

Step 23

Simplify the expression.

Tap for more steps...

Step 23.1

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

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

Step 23.2

Move $3$ to the left of $x^{2} - 2$ .

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

Step 24

Multiply $\frac{1}{x {(x^{2} - 2)}^{\frac{2}{3}}}$ by $\frac{4 x^{2} + 3 (x^{2} - 2)}{3 {(x^{2} - 2)}^{\frac{1}{3}}}$ .

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

Step 25

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

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

Step 26

Simplify the expression.

Tap for more steps...

Step 26.1

Combine the numerators over the common denominator.

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

Step 26.2

Add $2$ and $1$ .

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

Step 27

Cancel the common factor of $3$ .

Tap for more steps...

Step 27.1

Cancel the common factor.

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

Step 27.2

Rewrite the expression.

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

Step 28

Simplify.

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

Step 29

Simplify.

Tap for more steps...

Step 29.1

Apply the distributive property.

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

Step 29.2

Apply the distributive property.

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

Step 29.3

Apply the distributive property.

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

Step 29.4

Simplify the numerator.

Tap for more steps...

Step 29.4.1

Multiply $3$ by $- 2$ .

$\frac{4 x^{2} + 3 x^{2} - 6}{x (3 x^{2}) + x (3 \cdot - 2)}$

Step 29.4.2

Add $4 x^{2}$ and $3 x^{2}$ .

$\frac{7 x^{2} - 6}{x (3 x^{2}) + x (3 \cdot - 2)}$

Step 29.5

Combine terms.

Tap for more steps...

Step 29.5.1

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

$\frac{7 x^{2} - 6}{3 (x^{1} x^{2}) + x (3 \cdot - 2)}$

Step 29.5.2

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

$\frac{7 x^{2} - 6}{3 x^{1 + 2} + x (3 \cdot - 2)}$

Step 29.5.3

Add $1$ and $2$ .

$\frac{7 x^{2} - 6}{3 x^{3} + x (3 \cdot - 2)}$

Step 29.5.4

Multiply $3$ by $- 2$ .

$\frac{7 x^{2} - 6}{3 x^{3} + x \cdot - 6}$

Step 29.5.5

Move $- 6$ to the left of $x$ .

$\frac{7 x^{2} - 6}{3 x^{3} - 6 x}$

Step 29.6

Factor $3 x$ out of $3 x^{3} - 6 x$ .

Tap for more steps...

Step 29.6.1

Factor $3 x$ out of $3 x^{3}$ .

$\frac{7 x^{2} - 6}{3 x (x^{2}) - 6 x}$

Step 29.6.2

Factor $3 x$ out of $- 6 x$ .

$\frac{7 x^{2} - 6}{3 x (x^{2}) + 3 x (- 2)}$

Step 29.6.3

Factor $3 x$ out of $3 x (x^{2}) + 3 x (- 2)$ .

$\frac{7 x^{2} - 6}{3 x (x^{2} - 2)}$