Calculus Examples

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

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 $f (x) = x^{3}$ and $g (x) = {(x^{2} + 3)}^{\frac{1}{3}}$ .

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

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 $g (x) = x^{2} + 3$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 8

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

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

Step 9

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

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

Step 10

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Step 11

Combine fractions.

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Step 11.1

Add $2 x$ and $0$ .

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

Raise $x$ to the power of $1$ .

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

Step 13

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

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

Step 14

Add $1$ and $3$ .

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

Step 15

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

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

Step 16

Move $3$ to the left of ${(x^{2} + 3)}^{\frac{1}{3}}$ .

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

Step 17

Combine $\frac{2 x^{4}}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$ and $3 {(x^{2} + 3)}^{\frac{1}{3}} x^{2}$ using a common denominator.

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Step 17.1

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

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

Step 17.2

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

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

Step 17.3

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

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

Step 17.4

Combine the numerators over the common denominator.

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

Step 18

Multiply $3$ by $3$ .

$\frac{2 x^{4} + 9 x^{2} {(x^{2} + 3)}^{\frac{1}{3}} {(x^{2} + 3)}^{\frac{2}{3}}}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 19

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

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Step 19.1

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

$\frac{2 x^{4} + 9 x^{2} ({(x^{2} + 3)}^{\frac{2}{3}} {(x^{2} + 3)}^{\frac{1}{3}})}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 19.2

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

$\frac{2 x^{4} + 9 x^{2} {(x^{2} + 3)}^{\frac{2}{3} + \frac{1}{3}}}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 19.3

Combine the numerators over the common denominator.

$\frac{2 x^{4} + 9 x^{2} {(x^{2} + 3)}^{\frac{2 + 1}{3}}}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 19.4

Add $2$ and $1$ .

$\frac{2 x^{4} + 9 x^{2} {(x^{2} + 3)}^{\frac{3}{3}}}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 19.5

Divide $3$ by $3$ .

$\frac{2 x^{4} + 9 x^{2} {(x^{2} + 3)}^{1}}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 20

Simplify $9 x^{2} {(x^{2} + 3)}^{1}$ .

$\frac{2 x^{4} + 9 x^{2} (x^{2} + 3)}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 21

Simplify.

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Step 21.1

Apply the distributive property.

$\frac{2 x^{4} + 9 x^{2} x^{2} + 9 x^{2} \cdot 3}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 21.2

Simplify the numerator.

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Step 21.2.1

Simplify each term.

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Step 21.2.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 21.2.1.1.1

Move $x^{2}$ .

$\frac{2 x^{4} + 9 (x^{2} x^{2}) + 9 x^{2} \cdot 3}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 21.2.1.1.2

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

$\frac{2 x^{4} + 9 x^{2 + 2} + 9 x^{2} \cdot 3}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 21.2.1.1.3

Add $2$ and $2$ .

$\frac{2 x^{4} + 9 x^{4} + 9 x^{2} \cdot 3}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 21.2.1.2

Multiply $3$ by $9$ .

$\frac{2 x^{4} + 9 x^{4} + 27 x^{2}}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 21.2.2

Add $2 x^{4}$ and $9 x^{4}$ .

$\frac{11 x^{4} + 27 x^{2}}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 21.3

Factor $x^{2}$ out of $11 x^{4} + 27 x^{2}$ .

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Step 21.3.1

Factor $x^{2}$ out of $11 x^{4}$ .

$\frac{x^{2} (11 x^{2}) + 27 x^{2}}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 21.3.2

Factor $x^{2}$ out of $27 x^{2}$ .

$\frac{x^{2} (11 x^{2}) + x^{2} \cdot 27}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$

Step 21.3.3

Factor $x^{2}$ out of $x^{2} (11 x^{2}) + x^{2} \cdot 27$ .

$\frac{x^{2} (11 x^{2} + 27)}{3 {(x^{2} + 3)}^{\frac{2}{3}}}$