Calculus Examples

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{x}$ as $x^{\frac{1}{9}}$ .

$\frac{d}{d x} [x^{\frac{1}{9}} - \frac{9}{x}]$

Step 2

By the Sum Rule, the derivative of $x^{\frac{1}{9}} - \frac{9}{x}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{9}}] + \frac{d}{d x} [- \frac{9}{x}]$ .

$\frac{d}{d x} [x^{\frac{1}{9}}] + \frac{d}{d x} [- \frac{9}{x}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{9}$ .

$\frac{1}{9} x^{\frac{1}{9} - 1} + \frac{d}{d x} [- \frac{9}{x}]$

Step 4

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

$\frac{1}{9} x^{\frac{1}{9} - 1 \cdot \frac{9}{9}} + \frac{d}{d x} [- \frac{9}{x}]$

Step 5

Combine $- 1$ and $\frac{9}{9}$ .

$\frac{1}{9} x^{\frac{1}{9} + \frac{- 1 \cdot 9}{9}} + \frac{d}{d x} [- \frac{9}{x}]$

Step 6

Combine the numerators over the common denominator.

$\frac{1}{9} x^{\frac{1 - 1 \cdot 9}{9}} + \frac{d}{d x} [- \frac{9}{x}]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $9$ .

$\frac{1}{9} x^{\frac{1 - 9}{9}} + \frac{d}{d x} [- \frac{9}{x}]$

Step 7.2

Subtract $9$ from $1$ .

$\frac{1}{9} x^{\frac{- 8}{9}} + \frac{d}{d x} [- \frac{9}{x}]$

Step 8

Move the negative in front of the fraction.

$\frac{1}{9} x^{- \frac{8}{9}} + \frac{d}{d x} [- \frac{9}{x}]$

Step 9

Since $- 9$ is constant with respect to $x$ , the derivative of $- \frac{9}{x}$ with respect to $x$ is $- 9 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{1}{9} x^{- \frac{8}{9}} - 9 \frac{d}{d x} [\frac{1}{x}]$

Step 10

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

$\frac{1}{9} x^{- \frac{8}{9}} - 9 \frac{d}{d x} [x^{- 1}]$

Step 11

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

$\frac{1}{9} x^{- \frac{8}{9}} - 9 (- x^{- 2})$

Step 12

Multiply $- 1$ by $- 9$ .

$\frac{1}{9} x^{- \frac{8}{9}} + 9 x^{- 2}$

Step 13

Simplify.

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Step 13.1

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

$\frac{1}{9} \cdot \frac{1}{x^{\frac{8}{9}}} + 9 x^{- 2}$

Step 13.2

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

$\frac{1}{9} \cdot \frac{1}{x^{\frac{8}{9}}} + 9 \frac{1}{x^{2}}$

Step 13.3

Combine terms.

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Step 13.3.1

Multiply $\frac{1}{9}$ by $\frac{1}{x^{\frac{8}{9}}}$ .

$\frac{1}{9 x^{\frac{8}{9}}} + 9 \frac{1}{x^{2}}$

Step 13.3.2

Combine $9$ and $\frac{1}{x^{2}}$ .

$\frac{1}{9 x^{\frac{8}{9}}} + \frac{9}{x^{2}}$

Step 13.4

Reorder terms.

$\frac{9}{x^{2}} + \frac{1}{9 x^{\frac{8}{9}}}$