Calculus Examples

$f (x) = x^{\frac{5}{3}} - x^{\frac{2}{3}}$

Step 1

By the Sum Rule, the derivative of $x^{\frac{5}{3}} - x^{\frac{2}{3}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{5}{3}}] + \frac{d}{d x} [- x^{\frac{2}{3}}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [x^{\frac{5}{3}}]$ .

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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 $n = \frac{5}{3}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $3$ .

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

Step 2.5.2

Subtract $3$ from $5$ .

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

Step 3

Evaluate $\frac{d}{d x} [- x^{\frac{2}{3}}]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{2}{3}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$\frac{5}{3} x^{\frac{2}{3}} - \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $3$ .

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

Step 3.6.2

Subtract $3$ from $2$ .

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

Step 3.7

Move the negative in front of the fraction.

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

Step 3.8

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

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

Step 3.9

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

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

Step 4

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

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