Calculus Examples

$\sqrt{3 x^{3}}$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Factor out $x^{2}$ .

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

Step 1.1.2

Reorder $3$ and $x^{2}$ .

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Factor $3$ out of $3 x$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

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

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

Step 7.2

Combine the numerators over the common denominator.

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

Step 7.3

Add $2$ and $1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Simplify the numerator.

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Step 13.1

Multiply $- 1$ by $2$ .

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

Step 13.2

Subtract $2$ from $3$ .

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

Step 14

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

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

Step 15

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

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

Step 16

Multiply $3^{\frac{1}{2}}$ by $3$ by adding the exponents.

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Step 16.1

Move $3$ .

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

Step 16.2

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

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Step 16.2.1

Raise $3$ to the power of $1$ .

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

Step 16.2.2

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

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

Step 16.3

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

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

Step 16.4

Combine the numerators over the common denominator.

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

Step 16.5

Add $2$ and $1$ .

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