Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $3$ .

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

Step 7

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

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