Calculus Examples

$f (x) = \frac{1}{\sqrt{x^{13}}}$

Step 1

Rewrite $\frac{d}{d x} [\frac{1}{\sqrt{x^{13}}}]$ as $\frac{d}{d x} [\frac{1}{x^{6} \sqrt{x}}]$ .

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Step 1.1

Rewrite $x^{13}$ as ${(x^{6})}^{2} x$ .

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Step 1.1.1

Factor out $x^{12}$ .

$\frac{d}{d x} [\frac{1}{\sqrt{x^{12} x}}]$

Step 1.1.2

Rewrite $x^{12}$ as ${(x^{6})}^{2}$ .

$\frac{d}{d x} [\frac{1}{\sqrt{{(x^{6})}^{2} x}}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [\frac{1}{x^{6} \sqrt{x}}]$

Step 2

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

$\frac{d}{d x} [\frac{1}{x^{6} x^{\frac{1}{2}}}]$

Step 3

Multiply $x^{6}$ 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.

$\frac{d}{d x} [\frac{1}{x^{6 + \frac{1}{2}}}]$

Step 3.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{d}{d x} [\frac{1}{x^{6 \cdot \frac{2}{2} + \frac{1}{2}}}]$

Step 3.3

Combine $6$ and $\frac{2}{2}$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{6 \cdot 2}{2} + \frac{1}{2}}}]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [\frac{1}{x^{\frac{6 \cdot 2 + 1}{2}}}]$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $6$ by $2$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{12 + 1}{2}}}]$

Step 3.5.2

Add $12$ and $1$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{13}{2}}}]$

Step 4

Apply basic rules of exponents.

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Step 4.1

Rewrite $\frac{1}{x^{\frac{13}{2}}}$ as ${(x^{\frac{13}{2}})}^{- 1}$ .

$\frac{d}{d x} [{(x^{\frac{13}{2}})}^{- 1}]$

Step 4.2

Multiply the exponents in ${(x^{\frac{13}{2}})}^{- 1}$ .

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Step 4.2.1

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

$\frac{d}{d x} [x^{\frac{13}{2} \cdot - 1}]$

Step 4.2.2

Multiply $\frac{13}{2} \cdot - 1$ .

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Step 4.2.2.1

Combine $\frac{13}{2}$ and $- 1$ .

$\frac{d}{d x} [x^{\frac{13 \cdot - 1}{2}}]$

Step 4.2.2.2

Multiply $13$ by $- 1$ .

$\frac{d}{d x} [x^{\frac{- 13}{2}}]$

Step 4.2.3

Move the negative in front of the fraction.

$\frac{d}{d x} [x^{- \frac{13}{2}}]$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{13}{2}$ .

$- \frac{13}{2} x^{- \frac{13}{2} - 1}$

Step 6

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

$- \frac{13}{2} x^{- \frac{13}{2} - 1 \cdot \frac{2}{2}}$

Step 7

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

$- \frac{13}{2} x^{- \frac{13}{2} + \frac{- 1 \cdot 2}{2}}$

Step 8

Combine the numerators over the common denominator.

$- \frac{13}{2} x^{\frac{- 13 - 1 \cdot 2}{2}}$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$- \frac{13}{2} x^{\frac{- 13 - 2}{2}}$

Step 9.2

Subtract $2$ from $- 13$ .

$- \frac{13}{2} x^{\frac{- 15}{2}}$

Step 10

Move the negative in front of the fraction.

$- \frac{13}{2} x^{- \frac{15}{2}}$

Step 11

Simplify.

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Step 11.1

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

$- \frac{13}{2} \cdot \frac{1}{x^{\frac{15}{2}}}$

Step 11.2

Combine terms.

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Step 11.2.1

Multiply $\frac{1}{x^{\frac{15}{2}}}$ by $\frac{13}{2}$ .

$- \frac{13}{x^{\frac{15}{2}} \cdot 2}$

Step 11.2.2

Move $2$ to the left of $x^{\frac{15}{2}}$ .

$- \frac{13}{2 x^{\frac{15}{2}}}$