Calculus Examples

$y = \sqrt[3]{24 x^{2} + 8}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt[3]{24 x^{2} + 8}]$ as $\frac{d}{d x} [2 \sqrt[3]{3 x^{2} + 1}]$ .

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Step 1.1

Rewrite $24 x^{2} + 8$ as $2^{3} (3 x^{2} + 1)$ .

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Step 1.1.1

Factor $8$ out of $24 x^{2}$ .

$\frac{d}{d x} [\sqrt[3]{8 (3 x^{2}) + 8}]$

Step 1.1.2

Factor $8$ out of $8$ .

$\frac{d}{d x} [\sqrt[3]{8 (3 x^{2}) + 8 (1)}]$

Step 1.1.3

Factor $8$ out of $8 (3 x^{2}) + 8 (1)$ .

$\frac{d}{d x} [\sqrt[3]{8 (3 x^{2} + 1)}]$

Step 1.1.4

Rewrite $8$ as $2^{3}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

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

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

Step 2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 {(3 x^{2} + 1)}^{\frac{1}{3}}$ with respect to $x$ is $2 \frac{d}{d x} [{(3 x^{2} + 1)}^{\frac{1}{3}}]$ .

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

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u$ as $3 x^{2} + 1$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $3 x^{2} + 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

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

Step 7.2

Subtract $3$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

By the Sum Rule, the derivative of $3 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [1]$ .

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

Step 10

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 11

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

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

Step 12

Multiply $2$ by $3$ .

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

Step 13

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

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

Step 14

Simplify terms.

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Step 14.1

Add $6 x$ and $0$ .

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

Step 14.2

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

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

Step 14.3

Multiply $6$ by $2$ .

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

Step 14.4

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

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

Step 14.5

Factor $3$ out of $12 x$ .

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

Step 15

Cancel the common factors.

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Step 15.1

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

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

Step 15.2

Cancel the common factor.

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

Step 15.3

Rewrite the expression.

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