Calculus Examples

$y = \frac{3 \sqrt{x}}{x}$

Step 1

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

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

Step 2

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Subtract $1$ from $2$ .

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

Step 4

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x^{\frac{1}{2}}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $- 1$ .

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

Step 11

Move the negative in front of the fraction.

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

Step 12

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

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

Step 13

Multiply $- 1$ by $3$ .

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

Step 14

Combine $- 3$ and $\frac{x^{- \frac{3}{2}}}{2}$ .

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

Step 15

Simplify the expression.

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Step 15.1

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 15.2

Move the negative in front of the fraction.

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