Calculus Examples

$x^{\frac{20}{21}} - x^{\frac{41}{21}}$

Step 1

By the Sum Rule, the derivative of $x^{\frac{20}{21}} - x^{\frac{41}{21}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{20}{21}}] + \frac{d}{d x} [- x^{\frac{41}{21}}]$ .

$\frac{d}{d x} [x^{\frac{20}{21}}] + \frac{d}{d x} [- x^{\frac{41}{21}}]$

Step 2

Evaluate $\frac{d}{d x} [x^{\frac{20}{21}}]$ .

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{20}{21}$ .

$\frac{20}{21} x^{\frac{20}{21} - 1} + \frac{d}{d x} [- x^{\frac{41}{21}}]$

Step 2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$\frac{20}{21} x^{\frac{20}{21} - 1 \cdot \frac{21}{21}} + \frac{d}{d x} [- x^{\frac{41}{21}}]$

Step 2.3

Combine $- 1$ and $\frac{21}{21}$ .

$\frac{20}{21} x^{\frac{20}{21} + \frac{- 1 \cdot 21}{21}} + \frac{d}{d x} [- x^{\frac{41}{21}}]$

Step 2.4

Combine the numerators over the common denominator.

$\frac{20}{21} x^{\frac{20 - 1 \cdot 21}{21}} + \frac{d}{d x} [- x^{\frac{41}{21}}]$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $21$ .

$\frac{20}{21} x^{\frac{20 - 21}{21}} + \frac{d}{d x} [- x^{\frac{41}{21}}]$

Step 2.5.2

Subtract $21$ from $20$ .

$\frac{20}{21} x^{\frac{- 1}{21}} + \frac{d}{d x} [- x^{\frac{41}{21}}]$

Step 2.6

Move the negative in front of the fraction.

$\frac{20}{21} x^{- \frac{1}{21}} + \frac{d}{d x} [- x^{\frac{41}{21}}]$

Step 3

Evaluate $\frac{d}{d x} [- x^{\frac{41}{21}}]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{41}{21}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{41}{21}}]$ .

$\frac{20}{21} x^{- \frac{1}{21}} - \frac{d}{d x} [x^{\frac{41}{21}}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{41}{21}$ .

$\frac{20}{21} x^{- \frac{1}{21}} - (\frac{41}{21} x^{\frac{41}{21} - 1})$

Step 3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$\frac{20}{21} x^{- \frac{1}{21}} - (\frac{41}{21} x^{\frac{41}{21} - 1 \cdot \frac{21}{21}})$

Step 3.4

Combine $- 1$ and $\frac{21}{21}$ .

$\frac{20}{21} x^{- \frac{1}{21}} - (\frac{41}{21} x^{\frac{41}{21} + \frac{- 1 \cdot 21}{21}})$

Step 3.5

Combine the numerators over the common denominator.

$\frac{20}{21} x^{- \frac{1}{21}} - (\frac{41}{21} x^{\frac{41 - 1 \cdot 21}{21}})$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $21$ .

$\frac{20}{21} x^{- \frac{1}{21}} - (\frac{41}{21} x^{\frac{41 - 21}{21}})$

Step 3.6.2

Subtract $21$ from $41$ .

$\frac{20}{21} x^{- \frac{1}{21}} - (\frac{41}{21} x^{\frac{20}{21}})$

Step 3.7

Combine $\frac{41}{21}$ and $x^{\frac{20}{21}}$ .

$\frac{20}{21} x^{- \frac{1}{21}} - \frac{41 x^{\frac{20}{21}}}{21}$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{20}{21} \cdot \frac{1}{x^{\frac{1}{21}}} - \frac{41 x^{\frac{20}{21}}}{21}$

Step 4.2

Multiply $\frac{20}{21}$ by $\frac{1}{x^{\frac{1}{21}}}$ .

$\frac{20}{21 x^{\frac{1}{21}}} - \frac{41 x^{\frac{20}{21}}}{21}$