Calculus Examples

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

Step 1

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = 4 x - 2$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $4 x - 2$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $4 x - 2$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

Move ${(4 x - 2)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7.4

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

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

Step 8

By the Sum Rule, the derivative of $4 x - 2$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$ .

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

Step 9

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 10

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

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

Step 11

Multiply $4$ by $1$ .

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

Step 12

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

$\frac{x}{3 {(4 x - 2)}^{\frac{2}{3}}} (4 + 0) + {(4 x - 2)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 13

Combine fractions.

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Step 13.1

Add $4$ and $0$ .

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

Step 13.2

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

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

Step 13.3

Move $4$ to the left of $x$ .

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

Step 14

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

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

Step 15

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

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

Step 16

To write ${(4 x - 2)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(4 x - 2)}^{\frac{2}{3}}}{3 {(4 x - 2)}^{\frac{2}{3}}}$ .

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

Step 17

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

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

Step 18

Combine the numerators over the common denominator.

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

Step 19

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

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Step 19.1

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

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

Step 19.2

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

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

Step 19.3

Combine the numerators over the common denominator.

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

Step 19.4

Add $2$ and $1$ .

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

Step 19.5

Divide $3$ by $3$ .

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

Step 20

Simplify ${(4 x - 2)}^{1} \cdot 3$ .

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

Step 21

Move $3$ to the left of $4 x - 2$ .

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

Step 22

Simplify.

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Step 22.1

Apply the distributive property.

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

Step 22.2

Simplify the numerator.

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Step 22.2.1

Simplify each term.

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Step 22.2.1.1

Multiply $4$ by $3$ .

$\frac{4 x + 12 x + 3 \cdot - 2}{3 {(4 x - 2)}^{\frac{2}{3}}}$

Step 22.2.1.2

Multiply $3$ by $- 2$ .

$\frac{4 x + 12 x - 6}{3 {(4 x - 2)}^{\frac{2}{3}}}$

Step 22.2.2

Add $4 x$ and $12 x$ .

$\frac{16 x - 6}{3 {(4 x - 2)}^{\frac{2}{3}}}$

Step 22.3

Factor $2$ out of $16 x - 6$ .

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Step 22.3.1

Factor $2$ out of $16 x$ .

$\frac{2 (8 x) - 6}{3 {(4 x - 2)}^{\frac{2}{3}}}$

Step 22.3.2

Factor $2$ out of $- 6$ .

$\frac{2 (8 x) + 2 (- 3)}{3 {(4 x - 2)}^{\frac{2}{3}}}$

Step 22.3.3

Factor $2$ out of $2 (8 x) + 2 (- 3)$ .

$\frac{2 (8 x - 3)}{3 {(4 x - 2)}^{\frac{2}{3}}}$