Calculus Examples

$x^{\frac{2}{3}} (20 - x)$

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 $g (x) = 20 - x$ .

$x^{\frac{2}{3}} \frac{d}{d x} [20 - x] + (20 - x) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $20 - x$ with respect to $x$ is $\frac{d}{d x} [20] + \frac{d}{d x} [- x]$ .

$x^{\frac{2}{3}} (\frac{d}{d x} [20] + \frac{d}{d x} [- x]) + (20 - x) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.2

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$x^{\frac{2}{3}} (0 + \frac{d}{d x} [- x]) + (20 - x) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.3

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

$x^{\frac{2}{3}} \frac{d}{d x} [- x] + (20 - x) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.4

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

$x^{\frac{2}{3}} (- \frac{d}{d x} [x]) + (20 - x) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.5

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

$x^{\frac{2}{3}} (- 1 \cdot 1) + (20 - x) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $- 1$ by $1$ .

$x^{\frac{2}{3}} \cdot - 1 + (20 - x) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.6.2

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

$- 1 \cdot x^{\frac{2}{3}} + (20 - x) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.6.3

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

$- x^{\frac{2}{3}} + (20 - x) \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.7

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

$- x^{\frac{2}{3}} + (20 - x) (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 3

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

$- x^{\frac{2}{3}} + (20 - x) (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 4

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

$- x^{\frac{2}{3}} + (20 - x) (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 5

Combine the numerators over the common denominator.

$- x^{\frac{2}{3}} + (20 - x) (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

$- x^{\frac{2}{3}} + (20 - x) (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 6.2

Subtract $3$ from $2$ .

$- x^{\frac{2}{3}} + (20 - x) (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 7

Move the negative in front of the fraction.

$- x^{\frac{2}{3}} + (20 - x) (\frac{2}{3} x^{- \frac{1}{3}})$

Step 8

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

$- x^{\frac{2}{3}} + (20 - x) \frac{2 x^{- \frac{1}{3}}}{3}$

Step 9

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

$- x^{\frac{2}{3}} + (20 - x) \frac{2}{3 x^{\frac{1}{3}}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$- x^{\frac{2}{3}} + 20 \frac{2}{3 x^{\frac{1}{3}}} - x \frac{2}{3 x^{\frac{1}{3}}}$

Step 10.2

Combine terms.

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Step 10.2.1

Combine $20$ and $\frac{2}{3 x^{\frac{1}{3}}}$ .

$- x^{\frac{2}{3}} + \frac{20 \cdot 2}{3 x^{\frac{1}{3}}} - x \frac{2}{3 x^{\frac{1}{3}}}$

Step 10.2.2

Multiply $20$ by $2$ .

$- x^{\frac{2}{3}} + \frac{40}{3 x^{\frac{1}{3}}} - x \frac{2}{3 x^{\frac{1}{3}}}$

Step 10.2.3

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

$- x^{\frac{2}{3}} + \frac{40}{3 x^{\frac{1}{3}}} - \frac{2 x}{3 x^{\frac{1}{3}}}$

Step 10.2.4

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

$- x^{\frac{2}{3}} + \frac{40}{3 x^{\frac{1}{3}}} - \frac{2 x \cdot x^{- \frac{1}{3}}}{3}$

Step 10.2.5

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

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Step 10.2.5.1

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

$- x^{\frac{2}{3}} + \frac{40}{3 x^{\frac{1}{3}}} - \frac{2 (x^{- \frac{1}{3}} x)}{3}$

Step 10.2.5.2

Multiply $x^{- \frac{1}{3}}$ by $x$ .

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Step 10.2.5.2.1

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

$- x^{\frac{2}{3}} + \frac{40}{3 x^{\frac{1}{3}}} - \frac{2 (x^{- \frac{1}{3}} x^{1})}{3}$

Step 10.2.5.2.2

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

$- x^{\frac{2}{3}} + \frac{40}{3 x^{\frac{1}{3}}} - \frac{2 x^{- \frac{1}{3} + 1}}{3}$

Step 10.2.5.3

Write $1$ as a fraction with a common denominator.

$- x^{\frac{2}{3}} + \frac{40}{3 x^{\frac{1}{3}}} - \frac{2 x^{- \frac{1}{3} + \frac{3}{3}}}{3}$

Step 10.2.5.4

Combine the numerators over the common denominator.

$- x^{\frac{2}{3}} + \frac{40}{3 x^{\frac{1}{3}}} - \frac{2 x^{\frac{- 1 + 3}{3}}}{3}$

Step 10.2.5.5

Add $- 1$ and $3$ .

$- x^{\frac{2}{3}} + \frac{40}{3 x^{\frac{1}{3}}} - \frac{2 x^{\frac{2}{3}}}{3}$

Step 10.2.6

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

$- x^{\frac{2}{3}} \cdot \frac{3}{3} - \frac{2 x^{\frac{2}{3}}}{3} + \frac{40}{3 x^{\frac{1}{3}}}$

Step 10.2.7

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

$\frac{- x^{\frac{2}{3}} \cdot 3}{3} - \frac{2 x^{\frac{2}{3}}}{3} + \frac{40}{3 x^{\frac{1}{3}}}$

Step 10.2.8

Combine the numerators over the common denominator.

$\frac{- x^{\frac{2}{3}} \cdot 3 - 2 x^{\frac{2}{3}}}{3} + \frac{40}{3 x^{\frac{1}{3}}}$

Step 10.2.9

Multiply $3$ by $- 1$ .

$\frac{- 3 x^{\frac{2}{3}} - 2 x^{\frac{2}{3}}}{3} + \frac{40}{3 x^{\frac{1}{3}}}$

Step 10.2.10

Subtract $2 x^{\frac{2}{3}}$ from $- 3 x^{\frac{2}{3}}$ .

$\frac{- 5 x^{\frac{2}{3}}}{3} + \frac{40}{3 x^{\frac{1}{3}}}$

Step 10.2.11

Move the negative in front of the fraction.

$- \frac{5 x^{\frac{2}{3}}}{3} + \frac{40}{3 x^{\frac{1}{3}}}$

Step 10.3

Reorder terms.

$\frac{40}{3 x^{\frac{1}{3}}} - \frac{5 x^{\frac{2}{3}}}{3}$