Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

Subtract $2$ from $- 1$ .

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

Step 6

Move the negative in front of the fraction.

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

Step 7

Simplify.

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Step 7.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7.2

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

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

Step 7.3

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

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