Calculus Examples

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

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 $f (x) = x^{\frac{2}{3}}$ and $g (x) = {(6 - x)}^{\frac{1}{3}}$ .

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

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 $g (x) = 6 - x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $6 - x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $6 - x$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 8

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

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

Step 9

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{x^{\frac{2}{3}}}{3 {(6 - x)}^{\frac{2}{3}}} (0 + \frac{d}{d x} [- x]) + {(6 - x)}^{\frac{1}{3}} \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 10

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 11

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 12

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

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

Step 13

Combine fractions.

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Step 13.1

Multiply $- 1$ by $1$ .

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

Step 13.2

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

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

Step 13.3

Simplify the expression.

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Step 13.3.1

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

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

Step 13.3.2

Rewrite $- 1 x^{\frac{2}{3}}$ as $- x^{\frac{2}{3}}$ .

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

Step 13.3.3

Move the negative in front of the fraction.

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

Step 14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 15

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

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

Step 16

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

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

Step 17

Combine the numerators over the common denominator.

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

Step 18

Simplify the numerator.

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Step 18.1

Multiply $- 1$ by $3$ .

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

Step 18.2

Subtract $3$ from $2$ .

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

Step 19

Move the negative in front of the fraction.

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

Step 20

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

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

Step 21

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

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

Step 22

Simplify the expression.

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Step 22.1

Move $2$ to the left of ${(6 - x)}^{\frac{1}{3}}$ .

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

Step 22.2

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

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

Step 23

To write $- \frac{x^{\frac{2}{3}}}{3 {(6 - x)}^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

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

Step 24

To write $\frac{2 {(6 - x)}^{\frac{1}{3}}}{3 x^{\frac{1}{3}}}$ as a fraction with a common denominator, multiply by $\frac{{(6 - x)}^{\frac{2}{3}}}{{(6 - x)}^{\frac{2}{3}}}$ .

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

Step 25

Write each expression with a common denominator of $3 {(6 - x)}^{\frac{2}{3}} x^{\frac{1}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 25.1

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

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

Step 25.2

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

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

Step 25.3

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

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

Step 26

Combine the numerators over the common denominator.

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

Step 27

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

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Step 27.1

Move $x^{\frac{1}{3}}$ .

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

Step 27.2

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

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

Step 27.3

Combine the numerators over the common denominator.

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

Step 27.4

Add $1$ and $2$ .

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

Step 27.5

Divide $3$ by $3$ .

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

Step 28

Simplify $- x^{1}$ .

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

Step 29

Multiply ${(6 - x)}^{\frac{1}{3}}$ by ${(6 - x)}^{\frac{2}{3}}$ by adding the exponents.

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Step 29.1

Move ${(6 - x)}^{\frac{2}{3}}$ .

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

Step 29.2

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

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

Step 29.3

Combine the numerators over the common denominator.

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

Step 29.4

Add $2$ and $1$ .

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

Step 29.5

Divide $3$ by $3$ .

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

Step 30

Simplify $2 {(6 - x)}^{1}$ .

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

Step 31

Simplify.

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Step 31.1

Apply the distributive property.

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

Step 31.2

Simplify the numerator.

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Step 31.2.1

Simplify each term.

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Step 31.2.1.1

Multiply $2$ by $6$ .

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

Step 31.2.1.2

Multiply $- 1$ by $2$ .

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

Step 31.2.2

Subtract $2 x$ from $- x$ .

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

Step 31.3

Combine terms.

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Step 31.3.1

Factor $3$ out of $- 3 x$ .

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

Step 31.3.2

Factor $3$ out of $12$ .

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

Step 31.3.3

Factor $3$ out of $3 (- x) + 3 (4)$ .

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

Step 31.3.4

Cancel the common factors.

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Step 31.3.4.1

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

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

Step 31.3.4.2

Cancel the common factor.

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

Step 31.3.4.3

Rewrite the expression.

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

Step 31.4

Factor $- 1$ out of $- x$ .

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

Step 31.5

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 31.6

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

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

Step 31.7

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

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

Step 31.8

Move the negative in front of the fraction.

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