Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2

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

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

Step 1.2.3

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $- 1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Simplify.

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Step 8.1

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 8.2

Combine terms.

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Step 8.2.1

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

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

Step 8.2.2

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

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