Calculus Examples

$\sqrt{\frac{1}{x}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{1}{x}}$ as ${(\frac{1}{x})}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = \frac{1}{x}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $\frac{1}{x}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\frac{1}{x}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Simplify the expression.

Tap for more steps...

Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 8

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

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

Step 9

Combine fractions.

Tap for more steps...

Step 9.1

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

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

Step 9.2

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

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

Step 10

Simplify.

Tap for more steps...

Step 10.1

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 10.2

Combine terms.

Tap for more steps...

Step 10.2.1

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

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

Step 10.2.2

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

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

Step 10.2.3

Multiply $x^{2}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 10.2.3.1

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

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

Step 10.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10.2.3.3

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

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

Step 10.2.3.4

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

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

Step 10.2.3.5

Combine the numerators over the common denominator.

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

Step 10.2.3.6

Simplify the numerator.

Tap for more steps...

Step 10.2.3.6.1

Multiply $2$ by $2$ .

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

Step 10.2.3.6.2

Add $- 1$ and $4$ .

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