Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = 6 x^{2} + 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $6 x^{2} + 1$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $6 x^{2} + 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $3$ .

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

Step 5.2

Subtract $3$ from $1$ .

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

Step 6

Combine fractions.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

By the Sum Rule, the derivative of $6 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [1]$ .

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

Step 8

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

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

Step 9

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

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

Step 10

Multiply $2$ by $6$ .

$\frac{1}{3 {(6 x^{2} + 1)}^{\frac{2}{3}}} (12 x + \frac{d}{d x} [1])$

Step 11

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

$\frac{1}{3 {(6 x^{2} + 1)}^{\frac{2}{3}}} (12 x + 0)$

Step 12

Simplify terms.

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Step 12.1

Add $12 x$ and $0$ .

$\frac{1}{3 {(6 x^{2} + 1)}^{\frac{2}{3}}} (12 x)$

Step 12.2

Combine $12$ and $\frac{1}{3 {(6 x^{2} + 1)}^{\frac{2}{3}}}$ .

$\frac{12}{3 {(6 x^{2} + 1)}^{\frac{2}{3}}} x$

Step 12.3

Combine $\frac{12}{3 {(6 x^{2} + 1)}^{\frac{2}{3}}}$ and $x$ .

$\frac{12 x}{3 {(6 x^{2} + 1)}^{\frac{2}{3}}}$

Step 12.4

Factor $3$ out of $12 x$ .

$\frac{3 (4 x)}{3 {(6 x^{2} + 1)}^{\frac{2}{3}}}$

Step 13

Cancel the common factors.

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Step 13.1

Factor $3$ out of $3 {(6 x^{2} + 1)}^{\frac{2}{3}}$ .

$\frac{3 (4 x)}{3 ({(6 x^{2} + 1)}^{\frac{2}{3}})}$

Step 13.2

Cancel the common factor.

$\frac{3 (4 x)}{3 {(6 x^{2} + 1)}^{\frac{2}{3}}}$

Step 13.3

Rewrite the expression.

$\frac{4 x}{{(6 x^{2} + 1)}^{\frac{2}{3}}}$