Calculus Examples

$f (x) = x^{\frac{3}{5}} (4 - x)$

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 $f (x) = x^{\frac{3}{5}}$ and $g (x) = 4 - x$ .

$x^{\frac{3}{5}} \frac{d}{d x} [4 - x] + (4 - x) \frac{d}{d x} [x^{\frac{3}{5}}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 - x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x]$ .

$x^{\frac{3}{5}} (\frac{d}{d x} [4] + \frac{d}{d x} [- x]) + (4 - x) \frac{d}{d x} [x^{\frac{3}{5}}]$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$x^{\frac{3}{5}} (0 + \frac{d}{d x} [- x]) + (4 - x) \frac{d}{d x} [x^{\frac{3}{5}}]$

Step 2.3

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

$x^{\frac{3}{5}} \frac{d}{d x} [- x] + (4 - x) \frac{d}{d x} [x^{\frac{3}{5}}]$

Step 2.4

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

$x^{\frac{3}{5}} (- \frac{d}{d x} [x]) + (4 - x) \frac{d}{d x} [x^{\frac{3}{5}}]$

Step 2.5

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

$x^{\frac{3}{5}} (- 1 \cdot 1) + (4 - x) \frac{d}{d x} [x^{\frac{3}{5}}]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $- 1$ by $1$ .

$x^{\frac{3}{5}} \cdot - 1 + (4 - x) \frac{d}{d x} [x^{\frac{3}{5}}]$

Step 2.6.2

Move $- 1$ to the left of $x^{\frac{3}{5}}$ .

$- 1 \cdot x^{\frac{3}{5}} + (4 - x) \frac{d}{d x} [x^{\frac{3}{5}}]$

Step 2.6.3

Rewrite $- 1 x^{\frac{3}{5}}$ as $- x^{\frac{3}{5}}$ .

$- x^{\frac{3}{5}} + (4 - x) \frac{d}{d x} [x^{\frac{3}{5}}]$

Step 2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{5}$ .

$- x^{\frac{3}{5}} + (4 - x) (\frac{3}{5} x^{\frac{3}{5} - 1})$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- x^{\frac{3}{5}} + (4 - x) (\frac{3}{5} x^{\frac{3}{5} - 1 \cdot \frac{5}{5}})$

Step 4

Combine $- 1$ and $\frac{5}{5}$ .

$- x^{\frac{3}{5}} + (4 - x) (\frac{3}{5} x^{\frac{3}{5} + \frac{- 1 \cdot 5}{5}})$

Step 5

Combine the numerators over the common denominator.

$- x^{\frac{3}{5}} + (4 - x) (\frac{3}{5} x^{\frac{3 - 1 \cdot 5}{5}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $5$ .

$- x^{\frac{3}{5}} + (4 - x) (\frac{3}{5} x^{\frac{3 - 5}{5}})$

Step 6.2

Subtract $5$ from $3$ .

$- x^{\frac{3}{5}} + (4 - x) (\frac{3}{5} x^{\frac{- 2}{5}})$

Step 7

Move the negative in front of the fraction.

$- x^{\frac{3}{5}} + (4 - x) (\frac{3}{5} x^{- \frac{2}{5}})$

Step 8

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

$- x^{\frac{3}{5}} + (4 - x) \frac{3 x^{- \frac{2}{5}}}{5}$

Step 9

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

$- x^{\frac{3}{5}} + (4 - x) \frac{3}{5 x^{\frac{2}{5}}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$- x^{\frac{3}{5}} + 4 \frac{3}{5 x^{\frac{2}{5}}} - x \frac{3}{5 x^{\frac{2}{5}}}$

Step 10.2

Combine terms.

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Step 10.2.1

Combine $4$ and $\frac{3}{5 x^{\frac{2}{5}}}$ .

$- x^{\frac{3}{5}} + \frac{4 \cdot 3}{5 x^{\frac{2}{5}}} - x \frac{3}{5 x^{\frac{2}{5}}}$

Step 10.2.2

Multiply $4$ by $3$ .

$- x^{\frac{3}{5}} + \frac{12}{5 x^{\frac{2}{5}}} - x \frac{3}{5 x^{\frac{2}{5}}}$

Step 10.2.3

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

$- x^{\frac{3}{5}} + \frac{12}{5 x^{\frac{2}{5}}} - \frac{3 x}{5 x^{\frac{2}{5}}}$

Step 10.2.4

Move $x^{\frac{2}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$- x^{\frac{3}{5}} + \frac{12}{5 x^{\frac{2}{5}}} - \frac{3 x \cdot x^{- \frac{2}{5}}}{5}$

Step 10.2.5

Multiply $x$ by $x^{- \frac{2}{5}}$ by adding the exponents.

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Step 10.2.5.1

Move $x^{- \frac{2}{5}}$ .

$- x^{\frac{3}{5}} + \frac{12}{5 x^{\frac{2}{5}}} - \frac{3 (x^{- \frac{2}{5}} x)}{5}$

Step 10.2.5.2

Multiply $x^{- \frac{2}{5}}$ by $x$ .

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Step 10.2.5.2.1

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

$- x^{\frac{3}{5}} + \frac{12}{5 x^{\frac{2}{5}}} - \frac{3 (x^{- \frac{2}{5}} x^{1})}{5}$

Step 10.2.5.2.2

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

$- x^{\frac{3}{5}} + \frac{12}{5 x^{\frac{2}{5}}} - \frac{3 x^{- \frac{2}{5} + 1}}{5}$

Step 10.2.5.3

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

$- x^{\frac{3}{5}} + \frac{12}{5 x^{\frac{2}{5}}} - \frac{3 x^{- \frac{2}{5} + \frac{5}{5}}}{5}$

Step 10.2.5.4

Combine the numerators over the common denominator.

$- x^{\frac{3}{5}} + \frac{12}{5 x^{\frac{2}{5}}} - \frac{3 x^{\frac{- 2 + 5}{5}}}{5}$

Step 10.2.5.5

Add $- 2$ and $5$ .

$- x^{\frac{3}{5}} + \frac{12}{5 x^{\frac{2}{5}}} - \frac{3 x^{\frac{3}{5}}}{5}$

Step 10.2.6

To write $- x^{\frac{3}{5}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- x^{\frac{3}{5}} \cdot \frac{5}{5} - \frac{3 x^{\frac{3}{5}}}{5} + \frac{12}{5 x^{\frac{2}{5}}}$

Step 10.2.7

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

$\frac{- x^{\frac{3}{5}} \cdot 5}{5} - \frac{3 x^{\frac{3}{5}}}{5} + \frac{12}{5 x^{\frac{2}{5}}}$

Step 10.2.8

Combine the numerators over the common denominator.

$\frac{- x^{\frac{3}{5}} \cdot 5 - 3 x^{\frac{3}{5}}}{5} + \frac{12}{5 x^{\frac{2}{5}}}$

Step 10.2.9

Multiply $5$ by $- 1$ .

$\frac{- 5 x^{\frac{3}{5}} - 3 x^{\frac{3}{5}}}{5} + \frac{12}{5 x^{\frac{2}{5}}}$

Step 10.2.10

Subtract $3 x^{\frac{3}{5}}$ from $- 5 x^{\frac{3}{5}}$ .

$\frac{- 8 x^{\frac{3}{5}}}{5} + \frac{12}{5 x^{\frac{2}{5}}}$

Step 10.2.11

Move the negative in front of the fraction.

$- \frac{8 x^{\frac{3}{5}}}{5} + \frac{12}{5 x^{\frac{2}{5}}}$

Step 10.3

Reorder terms.

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{8 x^{\frac{3}{5}}}{5}$