Calculus Examples

$g (x) = \sqrt[3]{x^{5} + 6 x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{5} + 6 x}$ as ${(x^{5} + 6 x)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{(x^{5} + 6 x)}^{\frac{1}{3}}]$

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 $g (x) = x^{5} + 6 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{5} + 6 x$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [x^{5} + 6 x]$

Step 2.2

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

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [x^{5} + 6 x]$

Step 2.3

Replace all occurrences of $u$ with $x^{5} + 6 x$ .

$\frac{1}{3} {(x^{5} + 6 x)}^{\frac{1}{3} - 1} \frac{d}{d x} [x^{5} + 6 x]$

Step 3

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

$\frac{1}{3} {(x^{5} + 6 x)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x^{5} + 6 x]$

Step 4

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

$\frac{1}{3} {(x^{5} + 6 x)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x^{5} + 6 x]$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{3} {(x^{5} + 6 x)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [x^{5} + 6 x]$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(x^{5} + 6 x)}^{\frac{1 - 3}{3}} \frac{d}{d x} [x^{5} + 6 x]$

Step 6.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(x^{5} + 6 x)}^{\frac{- 2}{3}} \frac{d}{d x} [x^{5} + 6 x]$

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

$\frac{1}{3} {(x^{5} + 6 x)}^{- \frac{2}{3}} \frac{d}{d x} [x^{5} + 6 x]$

Step 7.2

Combine $\frac{1}{3}$ and ${(x^{5} + 6 x)}^{- \frac{2}{3}}$ .

$\frac{{(x^{5} + 6 x)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [x^{5} + 6 x]$

Step 7.3

Move ${(x^{5} + 6 x)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}} \frac{d}{d x} [x^{5} + 6 x]$

Step 8

By the Sum Rule, the derivative of $x^{5} + 6 x$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [6 x]$ .

$\frac{1}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}} (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [6 x])$

Step 9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}} (5 x^{4} + \frac{d}{d x} [6 x])$

Step 10

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

$\frac{1}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}} (5 x^{4} + 6 \frac{d}{d x} [x])$

Step 11

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

$\frac{1}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}} (5 x^{4} + 6 \cdot 1)$

Step 12

Multiply $6$ by $1$ .

$\frac{1}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}} (5 x^{4} + 6)$

Step 13

Simplify.

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Step 13.1

Reorder the factors of $\frac{1}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}} (5 x^{4} + 6)$ .

$(5 x^{4} + 6) \frac{1}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}}$

Step 13.2

Multiply $5 x^{4} + 6$ by $\frac{1}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}}$ .

$\frac{5 x^{4} + 6}{3 {(x^{5} + 6 x)}^{\frac{2}{3}}}$