Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $- 1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify the expression.

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Step 10.1

Multiply $2$ by $2$ .

$- \frac{x^{- \frac{3}{2}}}{4}$

Step 10.2

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

$- \frac{1}{4 x^{\frac{3}{2}}}$