Find dy First, divide dx square root of xy equals x to the power of two y plus one

$\sqrt{x y} = x^{2} y + 1$

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

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

Differentiate both sides of the equation.

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

Differentiate the left side of the equation.

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

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

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

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

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

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

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

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

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

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

Add numerators over the common denominator.

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

Clarify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Add fractions.

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Put the negative in front of the fraction.

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

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

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

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

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

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 $g (x) = y$ .

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

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

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

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

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

Multiply $y$ by $1$ .

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

Simplify.

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Apply the product rule to $x y$ .

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

Use the distributive law.

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

Combine terms.

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

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

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

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

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

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

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

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

Apply the power rule $a^{m} a^{n} = a^{m + n}$ to Add exponents.

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

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

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

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

Shift terms.

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

Differentiate the right side of the equation.

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By the Sum Rule, the derivative of $x^{2} y + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2} y] + \frac{d}{d x} [1]$ .

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

Evaluate $\frac{d}{d x} [x^{2} y]$ .

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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 $f (x) = x^{2}$ and $g (x) = y$ .

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

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

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

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

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

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

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

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$x^{2} \frac{d}{d x} [y] + 2 y x + 0$

Simplify.

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Add $x^{2} \frac{d}{d x} [y] + 2 y x$ and $0$ .

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

Shift terms.

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

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

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

Solve for $y'$ .

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

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

Subtract $x^{2} y'$ from both sides of the equation.

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

Calculate the lcd of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

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

Since $2 y^{\frac{1}{2}}, 2 x^{\frac{1}{2}}, 1, 1$ contain both numbers and variables, there are two steps to find the LCM. Calculate lcm for the numeric part $2, 2, 1, 1$ then find LCM for the variable part $y^{\frac{1}{2}}, x^{\frac{1}{2}}$ .

Lcm is considered the smallest positive number that all of the numbers divide into equally.

1. Write prime factors for all the numbers.

2. Each factor should be multiplied the biggest number of times it appears in each number.

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $2, 2, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

The LCM of $y^{\frac{1}{2}}, x^{\frac{1}{2}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

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

The LCM for $2 y^{\frac{1}{2}}, 2 x^{\frac{1}{2}}, 1, 1$ is the numeric part $2$ multiplied by the variable part.

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

Multiply each term by $2 y^{\frac{1}{2}} x^{\frac{1}{2}}$ and simplify.

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Multiply each term in $\frac{y' (x^{\frac{1}{2}})}{2 y^{\frac{1}{2}}} + \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - x^{2} y' = 2 x y$ by $2 y^{\frac{1}{2}} x^{\frac{1}{2}}$ in order to remove all the denominators from the equation.

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

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

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Clarify each term.

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Apply the commutative property rule to rewrite the expression.

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

The common factor should be canceled of $2$ .

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Cancel the common factor.

Reformulate the expression.

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

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

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

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

Cancel the common factor.

Rewrite the expression.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Simplify $y' x^{1}$ .

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

Rewrite using the commutative property of multiplication.

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

The common factor $2$ should be canceled.

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Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Simplify $y^{1}$ .

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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First, multiply $2$ by $2$ .

$y' x + y - x^{\frac{1 + 4}{2}} y' \cdot (2 y^{\frac{1}{2}}) = 2 x y \cdot (2 y^{\frac{1}{2}} x^{\frac{1}{2}})$

Add $1$ and $4$ .

$y' x + y - x^{\frac{5}{2}} y' \cdot (2 y^{\frac{1}{2}}) = 2 x y \cdot (2 y^{\frac{1}{2}} x^{\frac{1}{2}})$

Multiply $2$ by $- 1$ .

$y' x + y - 2 x^{\frac{5}{2}} y' y^{\frac{1}{2}} = 2 x y \cdot (2 y^{\frac{1}{2}} x^{\frac{1}{2}})$

Reorder factors in $y' x + y - 2 x^{\frac{5}{2}} y' y^{\frac{1}{2}}$ .

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x y \cdot (2 y^{\frac{1}{2}} x^{\frac{1}{2}})$

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

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

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

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 (x^{\frac{1}{2}} x) y \cdot (2 y^{\frac{1}{2}})$

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

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

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 (x^{\frac{1}{2}} x^{1}) y \cdot (2 y^{\frac{1}{2}})$

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

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{1}{2} + 1} y \cdot (2 y^{\frac{1}{2}})$

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

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{1}{2} + \frac{2}{2}} y \cdot (2 y^{\frac{1}{2}})$

Combine the numerators over the common denominator.

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{1 + 2}{2}} y \cdot (2 y^{\frac{1}{2}})$

Add $1$ and $2$ .

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{3}{2}} y \cdot (2 y^{\frac{1}{2}})$

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

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

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{3}{2}} (y^{\frac{1}{2}} y) \cdot 2$

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

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

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{3}{2}} (y^{\frac{1}{2}} y^{1}) \cdot 2$

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

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{3}{2}} y^{\frac{1}{2} + 1} \cdot 2$

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

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{3}{2}} y^{\frac{1}{2} + \frac{2}{2}} \cdot 2$

Combine the numerators over the common denominator.

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{3}{2}} y^{\frac{1 + 2}{2}} \cdot 2$

Add $1$ and $2$ .

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 2 x^{\frac{3}{2}} y^{\frac{3}{2}} \cdot 2$

Multiply $2$ by $2$ .

$y' x + y - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 4 x^{\frac{3}{2}} y^{\frac{3}{2}}$

Find solution to the equation.

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Subtract $y$ from both sides of the equation.

$y' x - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 4 x^{\frac{3}{2}} y^{\frac{3}{2}} - y$

Factor $y' x$ out of $y' x - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}}$ .

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Factor $y' x$ out of $y' x$ .

$y' (x) (1) - 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}} = 4 x^{\frac{3}{2}} y^{\frac{3}{2}} - y$

Factor $y' x$ out of $- 2 y' x^{\frac{5}{2}} y^{\frac{1}{2}}$ .

$y' (x) (1) + y' (x) (- 2 x^{\frac{3}{2}} y^{\frac{1}{2}}) = 4 x^{\frac{3}{2}} y^{\frac{3}{2}} - y$

Factor $y' x$ out of $y' x \cdot 1 + y' x (- 2 x^{\frac{3}{2}} y^{\frac{1}{2}})$ .

$y' (x) (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}) = 4 x^{\frac{3}{2}} y^{\frac{3}{2}} - y$

Divide each term by $x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})$ and simplify.

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Divide each term in $y' (x) (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}) = 4 x^{\frac{3}{2}} y^{\frac{3}{2}} - y$ by $x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})$ .

$\frac{y' (x) (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})} = \frac{4 x^{\frac{3}{2}} y^{\frac{3}{2}}}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Simplify $\frac{y' x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$ .

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Cancel the common factor.

$\frac{y' x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})} = \frac{4 x^{\frac{3}{2}} y^{\frac{3}{2}}}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Rewrite the expression.

$\frac{y' (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} = \frac{4 x^{\frac{3}{2}} y^{\frac{3}{2}}}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Cancel the common factor.

Divide $y'$ by $1$ .

$y' = \frac{4 x^{\frac{3}{2}} y^{\frac{3}{2}}}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Simplify $\frac{4 x^{\frac{3}{2}} y^{\frac{3}{2}}}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$ .

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Simplify each term.

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

$y' = \frac{4 x^{\frac{3}{2}} y^{\frac{3}{2}} x^{- 1}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Multiply $x^{\frac{3}{2}}$ by $x^{- 1}$ by adding the exponents.

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Move $x^{- 1}$ .

$y' = \frac{4 (x^{- 1} x^{\frac{3}{2}}) y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

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

$y' = \frac{4 x^{- 1 + \frac{3}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

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

$y' = \frac{4 x^{- 1 \cdot \frac{2}{2} + \frac{3}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

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

$y' = \frac{4 x^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Combine the numerators over the common denominator.

$y' = \frac{4 x^{\frac{- 1 \cdot 2 + 3}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

$y' = \frac{4 x^{\frac{- 2 + 3}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Add $- 2$ and $3$ .

$y' = \frac{4 x^{\frac{1}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} + \frac{- y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Move the negative in front of the fraction.

$y' = \frac{4 x^{\frac{1}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} - \frac{y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

To write $\frac{4 x^{\frac{1}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y' = \frac{4 x^{\frac{1}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}} \cdot \frac{x}{x} - \frac{y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Write each expression with a common denominator of $(1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}) x$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{4 x^{\frac{1}{2}} y^{\frac{3}{2}}}{1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}}$ and $\frac{x}{x}$ .

$y' = \frac{4 x^{\frac{1}{2}} y^{\frac{3}{2}} x}{(1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}) x} - \frac{y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Reorder the factors of $(1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}}) x$ .

$y' = \frac{4 x^{\frac{1}{2}} y^{\frac{3}{2}} x}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})} - \frac{y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Combine the numerators over the common denominator.

$y' = \frac{4 x^{\frac{1}{2}} y^{\frac{3}{2}} x - y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Simplify the numerator.

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Factor $y$ out of $4 x^{\frac{1}{2}} y^{\frac{3}{2}} x - y$ .

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Factor $y$ out of $4 x^{\frac{1}{2}} y^{\frac{3}{2}} x$ .

$y' = \frac{y (4 x^{\frac{1}{2}} y^{\frac{1}{2}} x) - y}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Factor $y$ out of $- y$ .

$y' = \frac{y (4 x^{\frac{1}{2}} y^{\frac{1}{2}} x) + y \cdot - 1}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Factor $y$ out of $y (4 x^{\frac{1}{2}} y^{\frac{1}{2}} x) + y \cdot - 1$ .

$y' = \frac{y (4 x^{\frac{1}{2}} y^{\frac{1}{2}} x - 1)}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

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

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Put $x$ .

$y' = \frac{y (4 (x \cdot x^{\frac{1}{2}}) y^{\frac{1}{2}} - 1)}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

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

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

$y' = \frac{y (4 (x^{1} x^{\frac{1}{2}}) y^{\frac{1}{2}} - 1)}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

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

$y' = \frac{y (4 x^{1 + \frac{1}{2}} y^{\frac{1}{2}} - 1)}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

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

$y' = \frac{y (4 x^{\frac{2}{2} + \frac{1}{2}} y^{\frac{1}{2}} - 1)}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Combine the numerators over the common denominator.

$y' = \frac{y (4 x^{\frac{2 + 1}{2}} y^{\frac{1}{2}} - 1)}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

Add $2$ and $1$ .

$y' = \frac{y (4 x^{\frac{3}{2}} y^{\frac{1}{2}} - 1)}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

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

$\frac{d y}{d x} = \frac{y (4 x^{\frac{3}{2}} y^{\frac{1}{2}} - 1)}{x (1 - 2 x^{\frac{3}{2}} y^{\frac{1}{2}})}$

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