Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Add $2$ and $1$ .

$\frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$

Step 3

Apply basic rules of exponents.

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Step 3.1

Rewrite $\frac{1}{x^{\frac{3}{2}}}$ as ${(x^{\frac{3}{2}})}^{- 1}$ .

$\frac{d}{d x} [{(x^{\frac{3}{2}})}^{- 1}]$

Step 3.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{- 1}$ .

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Step 3.2.1

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

$\frac{d}{d x} [x^{\frac{3}{2} \cdot - 1}]$

Step 3.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

$\frac{d}{d x} [x^{\frac{3 \cdot - 1}{2}}]$

Step 3.2.2.2

Multiply $3$ by $- 1$ .

$\frac{d}{d x} [x^{\frac{- 3}{2}}]$

Step 3.2.3

Move the negative in front of the fraction.

$\frac{d}{d x} [x^{- \frac{3}{2}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{3}{2}$ .

$- \frac{3}{2} x^{- \frac{3}{2} - 1}$

Step 5

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

$- \frac{3}{2} x^{- \frac{3}{2} - 1 \cdot \frac{2}{2}}$

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

$- \frac{3}{2} x^{\frac{- 3 - 1 \cdot 2}{2}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$- \frac{3}{2} x^{\frac{- 3 - 2}{2}}$

Step 8.2

Subtract $2$ from $- 3$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

Combine terms.

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Step 10.2.1

Multiply $\frac{1}{x^{\frac{5}{2}}}$ by $\frac{3}{2}$ .

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

Step 10.2.2

Move $2$ to the left of $x^{\frac{5}{2}}$ .

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