Calculus Examples

$y = \sqrt[5]{x} (3 - 9 x)$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$y = x^{\frac{1}{5}} (3 - 9 x)$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{\frac{1}{5}} (3 - 9 x))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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 $f (x) = x^{\frac{1}{5}}$ and $g (x) = 3 - 9 x$ .

$x^{\frac{1}{5}} \frac{d}{d x} [3 - 9 x] + (3 - 9 x) \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 4.2

Differentiate.

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Step 4.2.1

By the Sum Rule, the derivative of $3 - 9 x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 9 x]$ .

$x^{\frac{1}{5}} (\frac{d}{d x} [3] + \frac{d}{d x} [- 9 x]) + (3 - 9 x) \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 4.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$x^{\frac{1}{5}} (0 + \frac{d}{d x} [- 9 x]) + (3 - 9 x) \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 4.2.3

Add $0$ and $\frac{d}{d x} [- 9 x]$ .

$x^{\frac{1}{5}} \frac{d}{d x} [- 9 x] + (3 - 9 x) \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 4.2.4

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$x^{\frac{1}{5}} (- 9 \frac{d}{d x} [x]) + (3 - 9 x) \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 4.2.5

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

$x^{\frac{1}{5}} (- 9 \cdot 1) + (3 - 9 x) \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 4.2.6

Simplify the expression.

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Step 4.2.6.1

Multiply $- 9$ by $1$ .

$x^{\frac{1}{5}} \cdot - 9 + (3 - 9 x) \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 4.2.6.2

Move $- 9$ to the left of $x^{\frac{1}{5}}$ .

$- 9 \cdot x^{\frac{1}{5}} + (3 - 9 x) \frac{d}{d x} [x^{\frac{1}{5}}]$

Step 4.2.7

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

$- 9 x^{\frac{1}{5}} + (3 - 9 x) (\frac{1}{5} x^{\frac{1}{5} - 1})$

Step 4.3

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

$- 9 x^{\frac{1}{5}} + (3 - 9 x) (\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}})$

Step 4.4

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

$- 9 x^{\frac{1}{5}} + (3 - 9 x) (\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}})$

Step 4.5

Combine the numerators over the common denominator.

$- 9 x^{\frac{1}{5}} + (3 - 9 x) (\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}})$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $5$ .

$- 9 x^{\frac{1}{5}} + (3 - 9 x) (\frac{1}{5} x^{\frac{1 - 5}{5}})$

Step 4.6.2

Subtract $5$ from $1$ .

$- 9 x^{\frac{1}{5}} + (3 - 9 x) (\frac{1}{5} x^{\frac{- 4}{5}})$

Step 4.7

Move the negative in front of the fraction.

$- 9 x^{\frac{1}{5}} + (3 - 9 x) (\frac{1}{5} x^{- \frac{4}{5}})$

Step 4.8

Combine $\frac{1}{5}$ and $x^{- \frac{4}{5}}$ .

$- 9 x^{\frac{1}{5}} + (3 - 9 x) \frac{x^{- \frac{4}{5}}}{5}$

Step 4.9

Move $x^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 9 x^{\frac{1}{5}} + (3 - 9 x) \frac{1}{5 x^{\frac{4}{5}}}$

Step 4.10

Simplify.

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Step 4.10.1

Apply the distributive property.

$- 9 x^{\frac{1}{5}} + 3 \frac{1}{5 x^{\frac{4}{5}}} - 9 x \frac{1}{5 x^{\frac{4}{5}}}$

Step 4.10.2

Combine terms.

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Step 4.10.2.1

Combine $3$ and $\frac{1}{5 x^{\frac{4}{5}}}$ .

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} - 9 x \frac{1}{5 x^{\frac{4}{5}}}$

Step 4.10.2.2

Combine $- 9$ and $\frac{1}{5 x^{\frac{4}{5}}}$ .

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + x \frac{- 9}{5 x^{\frac{4}{5}}}$

Step 4.10.2.3

Combine $x$ and $\frac{- 9}{5 x^{\frac{4}{5}}}$ .

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + \frac{x \cdot - 9}{5 x^{\frac{4}{5}}}$

Step 4.10.2.4

Move $- 9$ to the left of $x$ .

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + \frac{- 9 \cdot x}{5 x^{\frac{4}{5}}}$

Step 4.10.2.5

Move $x^{\frac{4}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + \frac{- 9 x \cdot x^{- \frac{4}{5}}}{5}$

Step 4.10.2.6

Multiply $x$ by $x^{- \frac{4}{5}}$ by adding the exponents.

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Step 4.10.2.6.1

Move $x^{- \frac{4}{5}}$ .

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + \frac{- 9 (x^{- \frac{4}{5}} x)}{5}$

Step 4.10.2.6.2

Multiply $x^{- \frac{4}{5}}$ by $x$ .

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Step 4.10.2.6.2.1

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

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + \frac{- 9 (x^{- \frac{4}{5}} x^{1})}{5}$

Step 4.10.2.6.2.2

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

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + \frac{- 9 x^{- \frac{4}{5} + 1}}{5}$

Step 4.10.2.6.3

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

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + \frac{- 9 x^{- \frac{4}{5} + \frac{5}{5}}}{5}$

Step 4.10.2.6.4

Combine the numerators over the common denominator.

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + \frac{- 9 x^{\frac{- 4 + 5}{5}}}{5}$

Step 4.10.2.6.5

Add $- 4$ and $5$ .

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} + \frac{- 9 x^{\frac{1}{5}}}{5}$

Step 4.10.2.7

Move the negative in front of the fraction.

$- 9 x^{\frac{1}{5}} + \frac{3}{5 x^{\frac{4}{5}}} - \frac{9 x^{\frac{1}{5}}}{5}$

Step 4.10.2.8

To write $- 9 x^{\frac{1}{5}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 9 x^{\frac{1}{5}} \cdot \frac{5}{5} - \frac{9 x^{\frac{1}{5}}}{5} + \frac{3}{5 x^{\frac{4}{5}}}$

Step 4.10.2.9

Combine $- 9 x^{\frac{1}{5}}$ and $\frac{5}{5}$ .

$\frac{- 9 x^{\frac{1}{5}} \cdot 5}{5} - \frac{9 x^{\frac{1}{5}}}{5} + \frac{3}{5 x^{\frac{4}{5}}}$

Step 4.10.2.10

Combine the numerators over the common denominator.

$\frac{- 9 x^{\frac{1}{5}} \cdot 5 - 9 x^{\frac{1}{5}}}{5} + \frac{3}{5 x^{\frac{4}{5}}}$

Step 4.10.2.11

Multiply $5$ by $- 9$ .

$\frac{- 45 x^{\frac{1}{5}} - 9 x^{\frac{1}{5}}}{5} + \frac{3}{5 x^{\frac{4}{5}}}$

Step 4.10.2.12

Subtract $9 x^{\frac{1}{5}}$ from $- 45 x^{\frac{1}{5}}$ .

$\frac{- 54 x^{\frac{1}{5}}}{5} + \frac{3}{5 x^{\frac{4}{5}}}$

Step 4.10.2.13

Move the negative in front of the fraction.

$- \frac{54 x^{\frac{1}{5}}}{5} + \frac{3}{5 x^{\frac{4}{5}}}$

Step 4.10.3

Reorder terms.

$\frac{3}{5 x^{\frac{4}{5}}} - \frac{54 x^{\frac{1}{5}}}{5}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{3}{5 x^{\frac{4}{5}}} - \frac{54 x^{\frac{1}{5}}}{5}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3}{5 x^{\frac{4}{5}}} - \frac{54 x^{\frac{1}{5}}}{5}$