Calculus Examples

$- \sqrt{x y} = x - 20 y$

Step 1

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

$- {(x y)}^{\frac{1}{2}} = x - 20 y$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (- {(x y)}^{\frac{1}{2}}) = \frac{d}{d x} (x - 20 y)$

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

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

Step 3.2

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 $g (x) = x y$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x y$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $x y$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.6.2

Subtract $2$ from $1$ .

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

Step 3.7

Combine fractions.

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Step 3.7.1

Move the negative in front of the fraction.

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.8

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

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

Step 3.9

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

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

Step 3.10

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

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

Step 3.11

Multiply $y$ by $1$ .

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

Step 3.12

Simplify.

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Step 3.12.1

Apply the product rule to $x y$ .

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

Step 3.12.2

Apply the distributive property.

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

Step 3.12.3

Combine terms.

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Step 3.12.3.1

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

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

Step 3.12.3.2

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

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

Step 3.12.3.3

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

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

Step 3.12.3.4

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

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Step 3.12.3.4.1

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

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

Step 3.12.3.4.2

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

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Step 3.12.3.4.2.1

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

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

Step 3.12.3.4.2.2

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

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

Step 3.12.3.4.3

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

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

Step 3.12.3.4.4

Combine the numerators over the common denominator.

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

Step 3.12.3.4.5

Add $- 1$ and $2$ .

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

Step 3.12.3.5

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

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

Step 3.12.3.6

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

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

Step 3.12.3.7

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

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Step 3.12.3.7.1

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

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Step 3.12.3.7.1.1

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

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

Step 3.12.3.7.1.2

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

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

Step 3.12.3.7.2

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

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

Step 3.12.3.7.3

Combine the numerators over the common denominator.

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

Step 3.12.3.7.4

Subtract $1$ from $2$ .

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

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $x - 20 y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 20 y]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 20 y]$

Step 4.1.2

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

$1 + \frac{d}{d x} [- 20 y]$

Step 4.2

Evaluate $\frac{d}{d x} [- 20 y]$ .

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Step 4.2.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 y$ with respect to $x$ is $- 20 \frac{d}{d x} [y]$ .

$1 - 20 \frac{d}{d x} [y]$

Step 4.2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$1 - 20 y'$

Step 5

Reform the equation by setting the left side equal to the right side.

$- \frac{y' (x^{\frac{1}{2}})}{2 y^{\frac{1}{2}}} - \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} = 1 - 20 y'$

Step 6

Solve for $y'$ .

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Step 6.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 6.1.1

Add $\frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ to both sides of the equation.

$- \frac{y' x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} - 1 + 20 y' = \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 6.1.2

Add $1$ to both sides of the equation.

$- \frac{y' x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20 y' = \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1$

Step 6.2

Factor $y'$ out of $- \frac{y' x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20 y'$ .

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Step 6.2.1

Factor $y'$ out of $- \frac{y' x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}$ .

$y' (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + 20 y' = \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1$

Step 6.2.2

Factor $y'$ out of $20 y'$ .

$y' (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + y' \cdot 20 = \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1$

Step 6.2.3

Factor $y'$ out of $y' (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + y' \cdot 20$ .

$y' (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20) = \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1$

Step 6.3

Divide each term in $y' (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20) = \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1$ by $- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20$ and simplify.

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Step 6.3.1

Divide each term in $y' (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20) = \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1$ by $- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20$ .

$\frac{y' (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20)}{- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20} = \frac{\frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20} + \frac{1}{- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor.

Step 6.3.2.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}}{- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20} + \frac{1}{- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Combine the numerators over the common denominator.

$y' = \frac{\frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1}{- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20}$

Step 6.3.3.2

Multiply the numerator and denominator of the fraction by $2 x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

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Step 6.3.3.2.1

Multiply $\frac{\frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1}{- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20}$ by $\frac{2 x^{\frac{1}{2}} y^{\frac{1}{2}}}{2 x^{\frac{1}{2}} y^{\frac{1}{2}}}$ .

$y' = \frac{2 x^{\frac{1}{2}} y^{\frac{1}{2}}}{2 x^{\frac{1}{2}} y^{\frac{1}{2}}} \cdot \frac{\frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1}{- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20}$

Step 6.3.3.2.2

Combine.

$y' = \frac{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (\frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 1)}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 20)}$

Step 6.3.3.3

Apply the distributive property.

$y' = \frac{2 x^{\frac{1}{2}} y^{\frac{1}{2}} \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4

Simplify by cancelling.

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Step 6.3.3.4.1

Cancel the common factor of $2 x^{\frac{1}{2}}$ .

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Step 6.3.3.4.1.1

Factor $2 x^{\frac{1}{2}}$ out of $2 x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$y' = \frac{2 x^{\frac{1}{2}} (y^{\frac{1}{2}}) \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.1.2

Cancel the common factor.

Step 6.3.3.4.1.3

Rewrite the expression.

$y' = \frac{y^{\frac{1}{2}} y^{\frac{1}{2}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.2

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

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Step 6.3.3.4.2.1

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

$y' = \frac{y^{\frac{1}{2} + \frac{1}{2}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.2.2

Combine the numerators over the common denominator.

$y' = \frac{y^{\frac{1 + 1}{2}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.2.3

Add $1$ and $1$ .

$y' = \frac{y^{\frac{2}{2}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.2.4

Divide $2$ by $2$ .

$y' = \frac{y^{1} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.3

Simplify $y^{1}$ .

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}) + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.4

Cancel the common factor of $2 y^{\frac{1}{2}}$ .

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Step 6.3.3.4.4.1

Move the leading negative in $- \frac{x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}}$ into the numerator.

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 x^{\frac{1}{2}} y^{\frac{1}{2}} \frac{- x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.4.2

Factor $2 y^{\frac{1}{2}}$ out of $2 x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 y^{\frac{1}{2}} (x^{\frac{1}{2}}) \frac{- x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.4.3

Cancel the common factor.

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{2 y^{\frac{1}{2}} x^{\frac{1}{2}} \frac{- x^{\frac{1}{2}}}{2 y^{\frac{1}{2}}} + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.4.4

Rewrite the expression.

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{x^{\frac{1}{2}} (- x^{\frac{1}{2}}) + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.5

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

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Step 6.3.3.4.5.1

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

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{x^{\frac{1}{2}} x^{\frac{1}{2}} \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.5.2

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

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{x^{\frac{1}{2} + \frac{1}{2}} \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.5.3

Combine the numerators over the common denominator.

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{x^{\frac{1 + 1}{2}} \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.5.4

Add $1$ and $1$ .

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{x^{\frac{2}{2}} \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.5.5

Divide $2$ by $2$ .

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{x^{1} \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.4.6

Simplify $x^{1} \cdot - 1$ .

$y' = \frac{y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1}{x \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.5

Simplify the numerator.

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Step 6.3.3.5.1

Factor $y^{\frac{1}{2}}$ out of $y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 1$ .

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Step 6.3.3.5.1.1

Reorder the expression.

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Step 6.3.3.5.1.1.1

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

$y' = \frac{y + 2 x^{\frac{1}{2}} \cdot 1 y^{\frac{1}{2}}}{x \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.5.1.1.2

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

$y' = \frac{y + 2 \cdot 1 x^{\frac{1}{2}} y^{\frac{1}{2}}}{x \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.5.1.2

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

$y' = \frac{y^{1} + 2 \cdot 1 x^{\frac{1}{2}} y^{\frac{1}{2}}}{x \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.5.1.3

Factor $y^{\frac{1}{2}}$ out of $y^{1}$ .

$y' = \frac{y^{\frac{1}{2}} y^{\frac{1}{2}} + 2 \cdot 1 x^{\frac{1}{2}} y^{\frac{1}{2}}}{x \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.5.1.4

Factor $y^{\frac{1}{2}}$ out of $2 \cdot 1 x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$y' = \frac{y^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} (2 \cdot 1 x^{\frac{1}{2}})}{x \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.5.1.5

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} (2 \cdot 1 x^{\frac{1}{2}})$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 \cdot 1 x^{\frac{1}{2}})}{x \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.5.2

Multiply $2$ by $1$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.6

Simplify the denominator.

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Step 6.3.3.6.1

Factor $x^{\frac{1}{2}}$ out of $x \cdot - 1 + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20$ .

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Step 6.3.3.6.1.1

Reorder the expression.

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Step 6.3.3.6.1.1.1

Reorder $x$ and $- 1$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{- 1 \cdot x + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 20}$

Step 6.3.3.6.1.1.2

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

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{- 1 \cdot x + 2 x^{\frac{1}{2}} \cdot 20 y^{\frac{1}{2}}}$

Step 6.3.3.6.1.1.3

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

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{- 1 \cdot x + 2 \cdot 20 x^{\frac{1}{2}} y^{\frac{1}{2}}}$

Step 6.3.3.6.1.2

Factor $x^{\frac{1}{2}}$ out of $- 1 \cdot x$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (- 1 \cdot x^{\frac{1}{2}}) + 2 \cdot 20 x^{\frac{1}{2}} y^{\frac{1}{2}}}$

Step 6.3.3.6.1.3

Factor $x^{\frac{1}{2}}$ out of $2 \cdot 20 x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (- 1 \cdot x^{\frac{1}{2}}) + x^{\frac{1}{2}} (2 \cdot 20 y^{\frac{1}{2}})}$

Step 6.3.3.6.1.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (- 1 \cdot x^{\frac{1}{2}}) + x^{\frac{1}{2}} (2 \cdot 20 y^{\frac{1}{2}})$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (- 1 \cdot x^{\frac{1}{2}} + 2 \cdot 20 y^{\frac{1}{2}})}$

Step 6.3.3.6.2

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

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (- x^{\frac{1}{2}} + 2 \cdot 20 y^{\frac{1}{2}})}$

Step 6.3.3.6.3

Multiply $2$ by $20$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (- x^{\frac{1}{2}} + 40 y^{\frac{1}{2}})}$

Step 6.3.3.7

Simplify with factoring out.

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Step 6.3.3.7.1

Factor $- 1$ out of $- x^{\frac{1}{2}}$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (- (x^{\frac{1}{2}}) + 40 y^{\frac{1}{2}})}$

Step 6.3.3.7.2

Factor $- 1$ out of $40 y^{\frac{1}{2}}$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (- (x^{\frac{1}{2}}) - (- 40 y^{\frac{1}{2}}))}$

Step 6.3.3.7.3

Factor $- 1$ out of $- (x^{\frac{1}{2}}) - (- 40 y^{\frac{1}{2}})$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (- (x^{\frac{1}{2}} - 40 y^{\frac{1}{2}}))}$

Step 6.3.3.7.4

Rewrite negatives.

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Step 6.3.3.7.4.1

Rewrite $- (x^{\frac{1}{2}} - 40 y^{\frac{1}{2}})$ as $- 1 (x^{\frac{1}{2}} - 40 y^{\frac{1}{2}})$ .

$y' = \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (- 1 (x^{\frac{1}{2}} - 40 y^{\frac{1}{2}}))}$

Step 6.3.3.7.4.2

Move the negative in front of the fraction.

$y' = - \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (x^{\frac{1}{2}} - 40 y^{\frac{1}{2}})}$

Step 7

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{y^{\frac{1}{2}} (y^{\frac{1}{2}} + 2 x^{\frac{1}{2}})}{x^{\frac{1}{2}} (x^{\frac{1}{2}} - 40 y^{\frac{1}{2}})}$