Calculus Examples

$f (x) = \sqrt{12 x}$

Step 1

Rewrite $\frac{d}{d x} [\sqrt{12 x}]$ as $\frac{d}{d x} [2 \sqrt{3 x}]$ .

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Step 1.1

Rewrite $12 x$ as $2^{2} \cdot (3 x)$ .

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Step 1.1.1

Factor $4$ out of $12$ .

$\frac{d}{d x} [\sqrt{4 (3) x}]$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.1.3

Add parentheses.

$\frac{d}{d x} [\sqrt{2^{2} \cdot (3 x)}]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [2 \sqrt{3 x}]$

Step 2

Simplify with factoring out.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

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

Step 2.2

Factor $3$ out of $3 x$ .

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

Step 2.3

Apply the product rule to $3 (x)$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

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

Step 8.2

Subtract $2$ from $1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify the expression.

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Step 13.1

Move $2$ to the left of $3^{\frac{1}{2}}$ .

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

Step 13.2

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

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

Step 14

Cancel the common factor.

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

Step 15

Rewrite the expression.

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