Calculus Examples

$\frac{1}{\sqrt{x^{9}}}$

Step 1

Rewrite $\frac{d}{d x} [\frac{1}{\sqrt{x^{9}}}]$ as $\frac{d}{d x} [\frac{1}{x^{4} \sqrt{x}}]$ .

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Step 1.1

Rewrite $x^{9}$ as ${(x^{4})}^{2} x$ .

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Step 1.1.1

Factor out $x^{8}$ .

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

Step 1.1.2

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$\frac{d}{d x} [\frac{1}{\sqrt{{(x^{4})}^{2} x}}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [\frac{1}{x^{4} \sqrt{x}}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{1}{x^{4} x^{\frac{1}{2}}}]$

Step 3

Multiply $x^{4}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 3.1

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

$\frac{d}{d x} [\frac{1}{x^{4 + \frac{1}{2}}}]$

Step 3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{d}{d x} [\frac{1}{x^{4 \cdot \frac{2}{2} + \frac{1}{2}}}]$

Step 3.3

Combine $4$ and $\frac{2}{2}$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{4 \cdot 2}{2} + \frac{1}{2}}}]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [\frac{1}{x^{\frac{4 \cdot 2 + 1}{2}}}]$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $4$ by $2$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{8 + 1}{2}}}]$

Step 3.5.2

Add $8$ and $1$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{9}{2}}}]$

Step 4

Apply basic rules of exponents.

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Step 4.1

Rewrite $\frac{1}{x^{\frac{9}{2}}}$ as ${(x^{\frac{9}{2}})}^{- 1}$ .

$\frac{d}{d x} [{(x^{\frac{9}{2}})}^{- 1}]$

Step 4.2

Multiply the exponents in ${(x^{\frac{9}{2}})}^{- 1}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [x^{\frac{9}{2} \cdot - 1}]$

Step 4.2.2

Multiply $\frac{9}{2} \cdot - 1$ .

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Step 4.2.2.1

Combine $\frac{9}{2}$ and $- 1$ .

$\frac{d}{d x} [x^{\frac{9 \cdot - 1}{2}}]$

Step 4.2.2.2

Multiply $9$ by $- 1$ .

$\frac{d}{d x} [x^{\frac{- 9}{2}}]$

Step 4.2.3

Move the negative in front of the fraction.

$\frac{d}{d x} [x^{- \frac{9}{2}}]$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{9}{2}$ .

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

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$- \frac{9}{2} x^{\frac{- 9 - 2}{2}}$

Step 9.2

Subtract $2$ from $- 9$ .

$- \frac{9}{2} x^{\frac{- 11}{2}}$

Step 10

Move the negative in front of the fraction.

$- \frac{9}{2} x^{- \frac{11}{2}}$

Step 11

Simplify.

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Step 11.1

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

$- \frac{9}{2} \cdot \frac{1}{x^{\frac{11}{2}}}$

Step 11.2

Combine terms.

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Step 11.2.1

Multiply $\frac{1}{x^{\frac{11}{2}}}$ by $\frac{9}{2}$ .

$- \frac{9}{x^{\frac{11}{2}} \cdot 2}$

Step 11.2.2

Move $2$ to the left of $x^{\frac{11}{2}}$ .

$- \frac{9}{2 x^{\frac{11}{2}}}$