Calculus Examples

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

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x^{2}$ and $g (y) = y$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x$ .

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

Step 3.2.3.2

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

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

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $x$ .

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.3

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

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Step 3.3.1

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

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Step 3.3.1.1

To apply the Chain Rule, set $u_{2}$ as $2 x + y$ .

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

Step 3.3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 3.3.1.3

Replace all occurrences of $u_{2}$ with $2 x + y$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

$x^{2} + 2 y x x' + e^{2 x + y} (2 x' + 1)$

Step 3.4

Reorder terms.

$x^{2} + 2 x y x' + e^{2 x + y} (2 x' + 1)$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y^{\frac{1}{2}}$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $2$ .

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

Step 4.7.2

Subtract $2$ from $1$ .

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

Step 4.8

Move the negative in front of the fraction.

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

Simplify.

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Step 4.13.1

Apply the distributive property.

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

Step 4.13.2

Combine terms.

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Step 4.13.2.1

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

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

Step 4.13.2.2

Cancel the common factor.

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

Step 4.13.2.3

Rewrite the expression.

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

Step 4.13.3

Reorder terms.

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

Step 5

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

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

Step 6

Solve for $x'$ .

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Step 6.1

Simplify $x^{2} + 2 x y x' + e^{2 x + y} (2 x' + 1)$ .

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Step 6.1.1

Simplify each term.

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Step 6.1.1.1

Apply the distributive property.

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

Step 6.1.1.2

Rewrite using the commutative property of multiplication.

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

Step 6.1.1.3

Multiply $e^{2 x + y}$ by $1$ .

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

Step 6.1.2

Reorder factors in $x^{2} + 2 x y x' + 2 e^{2 x + y} x' + e^{2 x + y}$ .

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

Step 6.2

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

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

Step 6.3

Subtract $2 x' y^{\frac{1}{2}}$ from both sides of the equation.

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

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

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

Step 6.4.3

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

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

Step 6.5

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

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Step 6.5.1

Factor $2 x'$ out of $2 x y x'$ .

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

Step 6.5.2

Factor $2 x'$ out of $2 x' e^{2 x + y}$ .

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

Step 6.5.3

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

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

Step 6.5.4

Factor $2 x'$ out of $2 x' (x y) + 2 x' e^{2 x + y}$ .

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

Step 6.5.5

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

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

Step 6.6

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

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

Step 6.7

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

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Step 6.7.1

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

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

Step 6.7.2

Simplify the left side.

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Step 6.7.2.1

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

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

Step 6.7.2.2

Cancel the common factors.

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Step 6.7.2.2.1

Factor $2$ out of $2 x y$ .

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

Step 6.7.2.2.2

Factor $2$ out of $2 e^{2 x + y}$ .

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

Step 6.7.2.2.3

Factor $2$ out of $2 (x y) + 2 (e^{2 x + y})$ .

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

Step 6.7.2.2.4

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

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

Step 6.7.2.2.5

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

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

Step 6.7.2.2.6

Cancel the common factor.

Step 6.7.2.2.7

Rewrite the expression.

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

Step 6.7.2.3

Cancel the common factor.

Step 6.7.2.4

Divide $x'$ by $1$ .

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

Step 6.7.3

Simplify the right side.

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Step 6.7.3.1

Combine the numerators over the common denominator.

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

Step 6.7.3.2

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

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

Step 6.7.3.3

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

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

Step 6.7.3.4

Combine the numerators over the common denominator.

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

Step 6.7.3.5

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

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

Step 6.7.3.6

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

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

Step 6.7.3.7

Combine the numerators over the common denominator.

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

Step 6.7.3.8

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

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

Step 6.7.3.9

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

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

Step 6.7.3.10

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

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

Step 6.7.3.11

Factor $- 1$ out of $x$ .

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

Step 6.7.3.12

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

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

Step 6.7.3.13

Simplify the expression.

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Step 6.7.3.13.1

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

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

Step 6.7.3.13.2

Move the negative in front of the fraction.

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

Step 6.7.3.14

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.7.3.15

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

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Step 6.7.3.15.1

Factor $2$ out of $2 x y$ .

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

Step 6.7.3.15.2

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

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

Step 6.7.3.15.3

Factor $2$ out of $2 e^{2 x + y}$ .

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

Step 6.7.3.15.4

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

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

Step 6.7.3.15.5

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

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

Step 6.7.3.16

Multiply $\frac{1}{2 (x y - y^{\frac{1}{2}} + e^{2 x + y})}$ by $\frac{x^{2} y^{\frac{1}{2}} + e^{2 x + y} y^{\frac{1}{2}} - x}{y^{\frac{1}{2}}}$ .

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

Step 6.7.3.17

Reorder factors in $- \frac{x^{2} y^{\frac{1}{2}} + e^{2 x + y} y^{\frac{1}{2}} - x}{2 (x y - y^{\frac{1}{2}} + e^{2 x + y}) y^{\frac{1}{2}}}$ .

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

Step 7

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

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