Calculus Examples

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

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 $f (x) = x$ and $g (x) = {(x^{2} + 5)}^{\frac{1}{3}}$ .

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

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 $g (x) = x^{2} + 5$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2} + 5$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x^{2} + 5$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 8

By the Sum Rule, the derivative of $x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]$ .

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

Step 9

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

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

Step 10

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{x}{3 {(x^{2} + 5)}^{\frac{2}{3}}} (2 x + 0) + {(x^{2} + 5)}^{\frac{1}{3}} \frac{d}{d x} [x]$

Step 11

Combine fractions.

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Step 11.1

Add $2 x$ and $0$ .

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Add $1$ and $1$ .

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

Step 16

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

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

Step 17

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

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

Step 18

To write ${(x^{2} + 5)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x^{2} + 5)}^{\frac{2}{3}}}{3 {(x^{2} + 5)}^{\frac{2}{3}}}$ .

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

Step 19

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

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

Step 20

Combine the numerators over the common denominator.

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

Step 21

Multiply ${(x^{2} + 5)}^{\frac{1}{3}}$ by ${(x^{2} + 5)}^{\frac{2}{3}}$ by adding the exponents.

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Step 21.1

Move ${(x^{2} + 5)}^{\frac{2}{3}}$ .

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

Step 21.2

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

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

Step 21.3

Combine the numerators over the common denominator.

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

Step 21.4

Add $2$ and $1$ .

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

Step 21.5

Divide $3$ by $3$ .

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

Step 22

Simplify ${(x^{2} + 5)}^{1} \cdot 3$ .

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

Step 23

Move $3$ to the left of $x^{2} + 5$ .

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

Step 24

Simplify.

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Step 24.1

Apply the distributive property.

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

Step 24.2

Simplify the numerator.

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Step 24.2.1

Multiply $3$ by $5$ .

$\frac{2 x^{2} + 3 x^{2} + 15}{3 {(x^{2} + 5)}^{\frac{2}{3}}}$

Step 24.2.2

Add $2 x^{2}$ and $3 x^{2}$ .

$\frac{5 x^{2} + 15}{3 {(x^{2} + 5)}^{\frac{2}{3}}}$

Step 24.3

Factor $5$ out of $5 x^{2} + 15$ .

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Step 24.3.1

Factor $5$ out of $5 x^{2}$ .

$\frac{5 (x^{2}) + 15}{3 {(x^{2} + 5)}^{\frac{2}{3}}}$

Step 24.3.2

Factor $5$ out of $15$ .

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

Step 24.3.3

Factor $5$ out of $5 x^{2} + 5 \cdot 3$ .

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