Calculus Examples

Find the derivative of $\sqrt[3]{6 x + 2}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6 x + 2}$ as ${(6 x + 2)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{(6 x + 2)}^{\frac{1}{3}}]$

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 $g (x) = 6 x + 2$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $6 x + 2$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [6 x + 2]$

Step 2.2

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

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [6 x + 2]$

Step 2.3

Replace all occurrences of $u$ with $6 x + 2$ .

$\frac{1}{3} {(6 x + 2)}^{\frac{1}{3} - 1} \frac{d}{d x} [6 x + 2]$

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(6 x + 2)}^{\frac{1 - 3}{3}} \frac{d}{d x} [6 x + 2]$

Step 6.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(6 x + 2)}^{\frac{- 2}{3}} \frac{d}{d x} [6 x + 2]$

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

$\frac{1}{3} {(6 x + 2)}^{- \frac{2}{3}} \frac{d}{d x} [6 x + 2]$

Step 7.2

Combine $\frac{1}{3}$ and ${(6 x + 2)}^{- \frac{2}{3}}$ .

$\frac{{(6 x + 2)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [6 x + 2]$

Step 7.3

Move ${(6 x + 2)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(6 x + 2)}^{\frac{2}{3}}} \frac{d}{d x} [6 x + 2]$

Step 8

By the Sum Rule, the derivative of $6 x + 2$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [2]$ .

$\frac{1}{3 {(6 x + 2)}^{\frac{2}{3}}} (\frac{d}{d x} [6 x] + \frac{d}{d x} [2])$

Step 9

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{1}{3 {(6 x + 2)}^{\frac{2}{3}}} (6 \frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 10

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

$\frac{1}{3 {(6 x + 2)}^{\frac{2}{3}}} (6 \cdot 1 + \frac{d}{d x} [2])$

Step 11

Multiply $6$ by $1$ .

$\frac{1}{3 {(6 x + 2)}^{\frac{2}{3}}} (6 + \frac{d}{d x} [2])$

Step 12

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

$\frac{1}{3 {(6 x + 2)}^{\frac{2}{3}}} (6 + 0)$

Step 13

Simplify terms.

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Step 13.1

Add $6$ and $0$ .

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

Step 13.2

Combine $\frac{1}{3 {(6 x + 2)}^{\frac{2}{3}}}$ and $6$ .

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

Step 13.3

Factor $3$ out of $6$ .

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

Step 14

Cancel the common factors.

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Step 14.1

Factor $3$ out of $3 {(6 x + 2)}^{\frac{2}{3}}$ .

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

Step 14.2

Cancel the common factor.

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

Step 14.3

Rewrite the expression.

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