Calculus Examples

$\sqrt{\frac{x}{y}}$

Step 1

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

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

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 $g (x) = \frac{x}{y}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

Combine fractions.

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Step 9.1

Multiply $\frac{1}{y}$ by $\frac{1}{2}$ .

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

Step 9.2

Move $2$ to the left of $y$ .

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

Step 10

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

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

Step 11

Multiply $\frac{1}{2 y}$ by $1$ .

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

Apply the product rule to $\frac{y}{x}$ .

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

Step 12.3

Combine terms.

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Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

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

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Step 12.3.3.1

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

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

Step 12.3.3.2

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

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Step 12.3.3.2.1

Raise $y$ to the power of $1$ .

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

Step 12.3.3.2.2

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

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

Step 12.3.3.3

Write $1$ as a fraction with a common denominator.

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

Step 12.3.3.4

Combine the numerators over the common denominator.

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

Step 12.3.3.5

Add $- 1$ and $2$ .

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

Step 12.4

Reorder terms.

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