Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $3$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Multiply $3$ by $2$ .

$\frac{6 x^{\frac{1}{2}}}{2}$

Step 10

Factor $2$ out of $6 x^{\frac{1}{2}}$ .

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

Step 11

Cancel the common factors.

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Step 11.1

Factor $2$ out of $2$ .

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

Step 11.2

Cancel the common factor.

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

Step 11.3

Rewrite the expression.

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

Step 11.4

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

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