Calculus Examples

$e^{x^{2} + y^{2}} = x e^{5 y} - y^{2} e^{\frac{5 x}{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (e^{x^{2} + y^{2}}) = \frac{d}{d x} (x e^{5 y} - y^{2} e^{\frac{5 x}{2}})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + y^{2}$ .

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

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 2.1.3

Replace all occurrences of $u_{1}$ with $x^{2} + y^{2}$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $x e^{5 y} - y^{2} e^{\frac{5 x}{2}}$ with respect to $x$ is $\frac{d}{d x} [x e^{5 y}] + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$ .

$\frac{d}{d x} [x e^{5 y}] + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$

Step 3.2

Evaluate $\frac{d}{d x} [x e^{5 y}]$ .

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Step 3.2.1

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 $g (x) = e^{5 y}$ .

$x \frac{d}{d x} [e^{5 y}] + e^{5 y} \frac{d}{d x} [x] + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$

Step 3.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 5 y$ .

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Step 3.2.2.1

To apply the Chain Rule, set $u_{1}$ as $5 y$ .

$x (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [5 y]) + e^{5 y} \frac{d}{d x} [x] + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$

Step 3.2.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$x (e^{u_{1}} \frac{d}{d x} [5 y]) + e^{5 y} \frac{d}{d x} [x] + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$

Step 3.2.2.3

Replace all occurrences of $u_{1}$ with $5 y$ .

$x (e^{5 y} \frac{d}{d x} [5 y]) + e^{5 y} \frac{d}{d x} [x] + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$

Step 3.2.3

Since $5$ is constant with respect to $x$ , the derivative of $5 y$ with respect to $x$ is $5 \frac{d}{d x} [y]$ .

$x (e^{5 y} (5 \frac{d}{d x} [y])) + e^{5 y} \frac{d}{d x} [x] + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$

Step 3.2.4

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

$x (e^{5 y} (5 y')) + e^{5 y} \frac{d}{d x} [x] + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$

Step 3.2.5

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

$x (e^{5 y} (5 y')) + e^{5 y} \cdot 1 + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$

Step 3.2.6

Multiply $e^{5 y}$ by $1$ .

$x e^{5 y} (5 y') + e^{5 y} + \frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$

Step 3.3

Evaluate $\frac{d}{d x} [- y^{2} e^{\frac{5 x}{2}}]$ .

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Step 3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- y^{2} e^{\frac{5 x}{2}}$ with respect to $x$ is $- \frac{d}{d x} [y^{2} e^{\frac{5 x}{2}}]$ .

$x e^{5 y} (5 y') + e^{5 y} - \frac{d}{d x} [y^{2} e^{\frac{5 x}{2}}]$

Step 3.3.2

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 $f (x) = y^{2}$ and $g (x) = e^{\frac{5 x}{2}}$ .

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} \frac{d}{d x} [e^{\frac{5 x}{2}}] + e^{\frac{5 x}{2}} \frac{d}{d x} [y^{2}])$

Step 3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = \frac{5 x}{2}$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u_{2}$ as $\frac{5 x}{2}$ .

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [\frac{5 x}{2}]) + e^{\frac{5 x}{2}} \frac{d}{d x} [y^{2}])$

Step 3.3.3.2

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

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} (e^{u_{2}} \frac{d}{d x} [\frac{5 x}{2}]) + e^{\frac{5 x}{2}} \frac{d}{d x} [y^{2}])$

Step 3.3.3.3

Replace all occurrences of $u_{2}$ with $\frac{5 x}{2}$ .

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} (e^{\frac{5 x}{2}} \frac{d}{d x} [\frac{5 x}{2}]) + e^{\frac{5 x}{2}} \frac{d}{d x} [y^{2}])$

Step 3.3.4

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5 x}{2}$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [x]$ .

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} (e^{\frac{5 x}{2}} (\frac{5}{2} \frac{d}{d x} [x])) + e^{\frac{5 x}{2}} \frac{d}{d x} [y^{2}])$

Step 3.3.5

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

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} (e^{\frac{5 x}{2}} (\frac{5}{2} \cdot 1)) + e^{\frac{5 x}{2}} \frac{d}{d x} [y^{2}])$

Step 3.3.6

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

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Step 3.3.6.1

To apply the Chain Rule, set $u_{3}$ as $y$ .

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} (e^{\frac{5 x}{2}} (\frac{5}{2} \cdot 1)) + e^{\frac{5 x}{2}} (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [y]))$

Step 3.3.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 2$ .

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} (e^{\frac{5 x}{2}} (\frac{5}{2} \cdot 1)) + e^{\frac{5 x}{2}} (2 u_{3} \frac{d}{d x} [y]))$

Step 3.3.6.3

Replace all occurrences of $u_{3}$ with $y$ .

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} (e^{\frac{5 x}{2}} (\frac{5}{2} \cdot 1)) + e^{\frac{5 x}{2}} (2 y \frac{d}{d x} [y]))$

Step 3.3.7

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

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

Step 3.3.8

Multiply $\frac{5}{2}$ by $1$ .

$x e^{5 y} (5 y') + e^{5 y} - (y^{2} (e^{\frac{5 x}{2}} \frac{5}{2}) + e^{\frac{5 x}{2}} (2 y y'))$

Step 3.3.9

Combine $e^{\frac{5 x}{2}}$ and $\frac{5}{2}$ .

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

Step 3.3.10

Move $5$ to the left of $e^{\frac{5 x}{2}}$ .

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

Step 3.3.11

Combine $y^{2}$ and $\frac{5 e^{\frac{5 x}{2}}}{2}$ .

$x e^{5 y} (5 y') + e^{5 y} - (\frac{y^{2} (5 e^{\frac{5 x}{2}})}{2} + e^{\frac{5 x}{2}} (2 y y'))$

Step 3.3.12

Move $5$ to the left of $y^{2}$ .

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

Step 3.3.13

Move $2$ to the left of $e^{\frac{5 x}{2}}$ .

$x e^{5 y} (5 y') + e^{5 y} - (\frac{5 y^{2} e^{\frac{5 x}{2}}}{2} + 2 e^{\frac{5 x}{2}} y y')$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$x e^{5 y} (5 y') + e^{5 y} - \frac{5 y^{2} e^{\frac{5 x}{2}}}{2} - (2 e^{\frac{5 x}{2}} y y')$

Step 3.4.2

Multiply $2$ by $- 1$ .

$x e^{5 y} (5 y') + e^{5 y} - \frac{5 y^{2} e^{\frac{5 x}{2}}}{2} - 2 e^{\frac{5 x}{2}} y y'$

Step 3.4.3

Reorder terms.

$- \frac{5 y^{2} e^{\frac{5 x}{2}}}{2} + 5 e^{5 y} x y' - 2 e^{\frac{5 x}{2}} y y' + e^{5 y}$

Step 3.4.4

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

$- \frac{5 y^{2} e^{\frac{5 x}{2}}}{2} + 5 x e^{5 y} y' - 2 y e^{\frac{5 x}{2}} y' + e^{5 y}$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Reorder.

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Step 5.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.2.2

Rewrite using the commutative property of multiplication.

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

Step 5.3.2.3

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

$- \frac{5 y^{2} e^{\frac{5 x}{2}}}{2} + 5 x y' e^{5 y} - 2 y y' e^{\frac{5 x}{2}} + e^{5 y} = 2 x e^{x^{2} + y^{2}} + 2 y y' e^{x^{2} + y^{2}}$

Step 5.4

Move all terms containing $y'$ to the left side of the equation.

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Step 5.4.1

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

$- \frac{5 y^{2} e^{\frac{5 x}{2}}}{2} + 5 x y' e^{5 y} - 2 y y' e^{\frac{5 x}{2}} + e^{5 y} - 2 y y' e^{x^{2} + y^{2}} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.4.4

Combine the numerators over the common denominator.

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

Step 5.4.5

Simplify the numerator.

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Step 5.4.5.1

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

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Step 5.4.5.1.1

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

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

Step 5.4.5.1.2

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

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

Step 5.4.5.1.3

Factor $y$ out of $y (- 5 y e^{\frac{5 x}{2}}) + y (- 2 y' e^{x^{2} + y^{2}} \cdot 2)$ .

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

Step 5.4.5.2

Multiply $2$ by $- 2$ .

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

Step 5.4.6

To write $5 x y' e^{5 y}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 2 y y' e^{\frac{5 x}{2}} + e^{5 y} + 5 x y' e^{5 y} \cdot \frac{2}{2} + \frac{y (- 5 y e^{\frac{5 x}{2}} - 4 y' e^{x^{2} + y^{2}})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.7

Combine $5 x y' e^{5 y}$ and $\frac{2}{2}$ .

$- 2 y y' e^{\frac{5 x}{2}} + e^{5 y} + \frac{5 x y' e^{5 y} \cdot 2}{2} + \frac{y (- 5 y e^{\frac{5 x}{2}} - 4 y' e^{x^{2} + y^{2}})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.8

Combine the numerators over the common denominator.

$- 2 y y' e^{\frac{5 x}{2}} + e^{5 y} + \frac{5 x y' e^{5 y} \cdot 2 + y (- 5 y e^{\frac{5 x}{2}} - 4 y' e^{x^{2} + y^{2}})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.9

Simplify the numerator.

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Step 5.4.9.1

Multiply $2$ by $5$ .

$- 2 y y' e^{\frac{5 x}{2}} + e^{5 y} + \frac{10 x y' e^{5 y} + y (- 5 y e^{\frac{5 x}{2}} - 4 y' e^{x^{2} + y^{2}})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.9.2

Apply the distributive property.

$- 2 y y' e^{\frac{5 x}{2}} + e^{5 y} + \frac{10 x y' e^{5 y} + y (- 5 y e^{\frac{5 x}{2}}) + y (- 4 y' e^{x^{2} + y^{2}})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.9.3

Rewrite using the commutative property of multiplication.

$- 2 y y' e^{\frac{5 x}{2}} + e^{5 y} + \frac{10 x y' e^{5 y} - 5 y (y e^{\frac{5 x}{2}}) + y (- 4 y' e^{x^{2} + y^{2}})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.9.4

Rewrite using the commutative property of multiplication.

$- 2 y y' e^{\frac{5 x}{2}} + e^{5 y} + \frac{10 x y' e^{5 y} - 5 y (y e^{\frac{5 x}{2}}) - 4 y (y' e^{x^{2} + y^{2}})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.9.5

Multiply $y$ by $y$ by adding the exponents.

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Step 5.4.9.5.1

Move $y$ .

$- 2 y y' e^{\frac{5 x}{2}} + e^{5 y} + \frac{10 x y' e^{5 y} - 5 (y \cdot y) e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.9.5.2

Multiply $y$ by $y$ .

$- 2 y y' e^{\frac{5 x}{2}} + e^{5 y} + \frac{10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.10

To write $- 2 y y' e^{\frac{5 x}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 2 y y' e^{\frac{5 x}{2}} \cdot \frac{2}{2} + \frac{10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}})}{2} + e^{5 y} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.11

Combine $- 2 y y' e^{\frac{5 x}{2}}$ and $\frac{2}{2}$ .

$\frac{- 2 y y' e^{\frac{5 x}{2}} \cdot 2}{2} + \frac{10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}})}{2} + e^{5 y} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.12

Combine the numerators over the common denominator.

$\frac{- 2 y y' e^{\frac{5 x}{2}} \cdot 2 + 10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}})}{2} + e^{5 y} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.13

Multiply $2$ by $- 2$ .

$\frac{- 4 y y' e^{\frac{5 x}{2}} + 10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}})}{2} + e^{5 y} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.14

To write $e^{5 y}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 4 y y' e^{\frac{5 x}{2}} + 10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}})}{2} + e^{5 y} \cdot \frac{2}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.15

Combine $e^{5 y}$ and $\frac{2}{2}$ .

$\frac{- 4 y y' e^{\frac{5 x}{2}} + 10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}})}{2} + \frac{e^{5 y} \cdot 2}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.16

Combine the numerators over the common denominator.

$\frac{- 4 y y' e^{\frac{5 x}{2}} + 10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}}) + e^{5 y} \cdot 2}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.17

Move $2$ to the left of $e^{5 y}$ .

$\frac{- 4 y y' e^{\frac{5 x}{2}} + 10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}}) + 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.18

Factor $- 1$ out of $- 4 y y' e^{\frac{5 x}{2}}$ .

$\frac{- (4 y y' e^{\frac{5 x}{2}}) + 10 x y' e^{5 y} - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}}) + 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.19

Factor $- 1$ out of $10 x y' e^{5 y}$ .

$\frac{- (4 y y' e^{\frac{5 x}{2}}) - (- 10 x y' e^{5 y}) - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}}) + 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.20

Factor $- 1$ out of $- (4 y y' e^{\frac{5 x}{2}}) - (- 10 x y' e^{5 y})$ .

$\frac{- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y}) - 5 y^{2} e^{\frac{5 x}{2}} - 4 y (y' e^{x^{2} + y^{2}}) + 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.21

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

$\frac{- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y}) - (5 y^{2} e^{\frac{5 x}{2}}) - 4 y (y' e^{x^{2} + y^{2}}) + 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.22

Factor $- 1$ out of $- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y}) - (5 y^{2} e^{\frac{5 x}{2}})$ .

$\frac{- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}}) - 4 y (y' e^{x^{2} + y^{2}}) + 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.23

Factor $- 1$ out of $- 4 y (y' e^{x^{2} + y^{2}})$ .

$\frac{- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}}) - (4 y (y' e^{x^{2} + y^{2}})) + 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.24

Factor $- 1$ out of $- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}}) - (4 y (y' e^{x^{2} + y^{2}}))$ .

$\frac{- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y (y' e^{x^{2} + y^{2}})) + 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.25

Factor $- 1$ out of $2 e^{5 y}$ .

$\frac{- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y (y' e^{x^{2} + y^{2}})) - (- 2 e^{5 y})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.26

Factor $- 1$ out of $- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y (y' e^{x^{2} + y^{2}})) - (- 2 e^{5 y})$ .

$\frac{- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y (y' e^{x^{2} + y^{2}}) - 2 e^{5 y})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.27

Rewrite $- (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y (y' e^{x^{2} + y^{2}}) - 2 e^{5 y})$ as $- 1 (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y (y' e^{x^{2} + y^{2}}) - 2 e^{5 y})$ .

$\frac{- 1 (4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y (y' e^{x^{2} + y^{2}}) - 2 e^{5 y})}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.28

Move the negative in front of the fraction.

$- \frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y (y' e^{x^{2} + y^{2}}) - 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.4.29

Reorder factors in $- \frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y (y' e^{x^{2} + y^{2}}) - 2 e^{5 y}}{2}$ .

$- \frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$

Step 5.5

Divide each term in $- \frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$ by $- 1$ and simplify.

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Step 5.5.1

Divide each term in $- \frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} = 2 x e^{x^{2} + y^{2}}$ by $- 1$ .

$\frac{- \frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2}}{- 1} = \frac{2 x e^{x^{2} + y^{2}}}{- 1}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2}}{1} = \frac{2 x e^{x^{2} + y^{2}}}{- 1}$

Step 5.5.2.2

Divide $\frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2}$ by $1$ .

$\frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} = \frac{2 x e^{x^{2} + y^{2}}}{- 1}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative one from the denominator of $\frac{2 x e^{x^{2} + y^{2}}}{- 1}$ .

$\frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} = - 1 \cdot (2 x e^{x^{2} + y^{2}})$

Step 5.5.3.2

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

$\frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} = - (2 x e^{x^{2} + y^{2}})$

Step 5.5.3.3

Multiply $2$ by $- 1$ .

$\frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} = - 2 x e^{x^{2} + y^{2}}$

Step 5.6

Multiply both sides by $2$ .

$\frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} \cdot 2 = - 2 x e^{x^{2} + y^{2}} \cdot 2$

Step 5.7

Simplify.

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Step 5.7.1

Simplify the left side.

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Step 5.7.1.1

Simplify $\frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} \cdot 2$ .

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Step 5.7.1.1.1

Cancel the common factor of $2$ .

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Step 5.7.1.1.1.1

Cancel the common factor.

$\frac{4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y}}{2} \cdot 2 = - 2 x e^{x^{2} + y^{2}} \cdot 2$

Step 5.7.1.1.1.2

Rewrite the expression.

$4 y y' e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y} = - 2 x e^{x^{2} + y^{2}} \cdot 2$

Step 5.7.1.1.2

Reorder.

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Step 5.7.1.1.2.1

Move $y$ .

$4 y' y e^{\frac{5 x}{2}} - 10 x y' e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y} = - 2 x e^{x^{2} + y^{2}} \cdot 2$

Step 5.7.1.1.2.2

Move $x$ .

$4 y' y e^{\frac{5 x}{2}} - 10 y' x e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y y' e^{x^{2} + y^{2}} - 2 e^{5 y} = - 2 x e^{x^{2} + y^{2}} \cdot 2$

Step 5.7.1.1.2.3

Move $y$ .

$4 y' y e^{\frac{5 x}{2}} - 10 y' x e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y' y e^{x^{2} + y^{2}} - 2 e^{5 y} = - 2 x e^{x^{2} + y^{2}} \cdot 2$

Step 5.7.1.1.2.4

Move $4 y' y e^{\frac{5 x}{2}}$ .

$- 10 y' x e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y' y e^{\frac{5 x}{2}} + 4 y' y e^{x^{2} + y^{2}} - 2 e^{5 y} = - 2 x e^{x^{2} + y^{2}} \cdot 2$

Step 5.7.2

Simplify the right side.

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Step 5.7.2.1

Multiply $2$ by $- 2$ .

$- 10 y' x e^{5 y} + 5 y^{2} e^{\frac{5 x}{2}} + 4 y' y e^{\frac{5 x}{2}} + 4 y' y e^{x^{2} + y^{2}} - 2 e^{5 y} = - 4 x e^{x^{2} + y^{2}}$

Step 5.8

Solve for $y'$ .

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Step 5.8.1

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

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Step 5.8.1.1

Subtract $5 y^{2} e^{\frac{5 x}{2}}$ from both sides of the equation.

$- 10 y' x e^{5 y} + 4 y' y e^{\frac{5 x}{2}} + 4 y' y e^{x^{2} + y^{2}} - 2 e^{5 y} = - 4 x e^{x^{2} + y^{2}} - 5 y^{2} e^{\frac{5 x}{2}}$

Step 5.8.1.2

Add $2 e^{5 y}$ to both sides of the equation.

$- 10 y' x e^{5 y} + 4 y' y e^{\frac{5 x}{2}} + 4 y' y e^{x^{2} + y^{2}} = - 4 x e^{x^{2} + y^{2}} - 5 y^{2} e^{\frac{5 x}{2}} + 2 e^{5 y}$

Step 5.8.2

Factor $2 y'$ out of $- 10 y' x e^{5 y} + 4 y' y e^{\frac{5 x}{2}} + 4 y' y e^{x^{2} + y^{2}}$ .

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Step 5.8.2.1

Factor $2 y'$ out of $- 10 y' x e^{5 y}$ .

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

Step 5.8.2.2

Factor $2 y'$ out of $4 y' y e^{\frac{5 x}{2}}$ .

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

Step 5.8.2.3

Factor $2 y'$ out of $4 y' y e^{x^{2} + y^{2}}$ .

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

Step 5.8.2.4

Factor $2 y'$ out of $2 y' (- 5 x e^{5 y}) + 2 y' (2 y e^{\frac{5 x}{2}})$ .

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

Step 5.8.2.5

Factor $2 y'$ out of $2 y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 2 y' (2 y e^{x^{2} + y^{2}})$ .

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

Step 5.8.3

Divide each term in $2 y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}) = - 4 x e^{x^{2} + y^{2}} - 5 y^{2} e^{\frac{5 x}{2}} + 2 e^{5 y}$ by $- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}$ and simplify.

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Step 5.8.3.1

$\frac{2 y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} = \frac{- 4 x e^{x^{2} + y^{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.2

Simplify the left side.

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Step 5.8.3.2.1

Cancel the common factor of $2$ and $- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}$ .

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Step 5.8.3.2.1.1

Factor $2$ out of $2 y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})$ .

$\frac{2 (y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}))}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} = \frac{- 4 x e^{x^{2} + y^{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.2.1.2

Cancel the common factors.

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Step 5.8.3.2.1.2.1

Factor $2$ out of $- 10 x e^{5 y}$ .

$\frac{2 (y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}))}{2 (- 5 x e^{5 y}) + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} = \frac{- 4 x e^{x^{2} + y^{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.2.1.2.2

Factor $2$ out of $4 y e^{\frac{5 x}{2}}$ .

$\frac{2 (y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}))}{2 (- 5 x e^{5 y}) + 2 (2 y e^{\frac{5 x}{2}}) + 4 y e^{x^{2} + y^{2}}} = \frac{- 4 x e^{x^{2} + y^{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.2.1.2.3

Factor $2$ out of $2 (- 5 x e^{5 y}) + 2 (2 y e^{\frac{5 x}{2}})$ .

$\frac{2 (y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}))}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 4 y e^{x^{2} + y^{2}}} = \frac{- 4 x e^{x^{2} + y^{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.2.1.2.4

Factor $2$ out of $4 y e^{x^{2} + y^{2}}$ .

$\frac{2 (y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}))}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 2 (2 y e^{x^{2} + y^{2}})} = \frac{- 4 x e^{x^{2} + y^{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.2.1.2.5

Factor $2$ out of $2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 2 (2 y e^{x^{2} + y^{2}})$ .

$\frac{2 (y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}))}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})} = \frac{- 4 x e^{x^{2} + y^{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.2.1.2.6

Cancel the common factor.

Step 5.8.3.2.1.2.7

Rewrite the expression.

$\frac{y' (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} = \frac{- 4 x e^{x^{2} + y^{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.2.2

Cancel the common factor of $- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}$ .

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Step 5.8.3.2.2.1

Cancel the common factor.

Step 5.8.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 4 x e^{x^{2} + y^{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3

Simplify the right side.

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Step 5.8.3.3.1

Simplify each term.

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Step 5.8.3.3.1.1

Cancel the common factor of $- 4$ and $- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}$ .

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Step 5.8.3.3.1.1.1

Factor $2$ out of $- 4 x e^{x^{2} + y^{2}}$ .

$y' = \frac{2 (- 2 x e^{x^{2} + y^{2}})}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.1.2

Cancel the common factors.

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Step 5.8.3.3.1.1.2.1

Factor $2$ out of $- 10 x e^{5 y}$ .

$y' = \frac{2 (- 2 x e^{x^{2} + y^{2}})}{2 (- 5 x e^{5 y}) + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.1.2.2

Factor $2$ out of $4 y e^{\frac{5 x}{2}}$ .

$y' = \frac{2 (- 2 x e^{x^{2} + y^{2}})}{2 (- 5 x e^{5 y}) + 2 (2 y e^{\frac{5 x}{2}}) + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.1.2.3

Factor $2$ out of $2 (- 5 x e^{5 y}) + 2 (2 y e^{\frac{5 x}{2}})$ .

$y' = \frac{2 (- 2 x e^{x^{2} + y^{2}})}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 4 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.1.2.4

Factor $2$ out of $4 y e^{x^{2} + y^{2}}$ .

$y' = \frac{2 (- 2 x e^{x^{2} + y^{2}})}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 2 (2 y e^{x^{2} + y^{2}})} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.1.2.5

Factor $2$ out of $2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 2 (2 y e^{x^{2} + y^{2}})$ .

$y' = \frac{2 (- 2 x e^{x^{2} + y^{2}})}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.1.2.6

Cancel the common factor.

Step 5.8.3.3.1.1.2.7

Rewrite the expression.

$y' = \frac{- 2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.3

Factor $2$ out of $- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}$ .

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Step 5.8.3.3.1.3.1

Factor $2$ out of $- 10 x e^{5 y}$ .

$y' = - \frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{2 (- 5 x e^{5 y}) + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.3.2

Factor $2$ out of $4 y e^{\frac{5 x}{2}}$ .

$y' = - \frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{2 (- 5 x e^{5 y}) + 2 (2 y e^{\frac{5 x}{2}}) + 4 y e^{x^{2} + y^{2}}} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.3.3

Factor $2$ out of $4 y e^{x^{2} + y^{2}}$ .

$y' = - \frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{2 (- 5 x e^{5 y}) + 2 (2 y e^{\frac{5 x}{2}}) + 2 (2 y e^{x^{2} + y^{2}})} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.3.4

Factor $2$ out of $2 (- 5 x e^{5 y}) + 2 (2 y e^{\frac{5 x}{2}})$ .

$y' = - \frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 2 (2 y e^{x^{2} + y^{2}})} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.3.5

Factor $2$ out of $2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 2 (2 y e^{x^{2} + y^{2}})$ .

$y' = - \frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} + \frac{- 5 y^{2} e^{\frac{5 x}{2}}}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} - \frac{5 y^{2} e^{\frac{5 x}{2}}}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})} + \frac{2 e^{5 y}}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.5

Cancel the common factor of $2$ and $- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}$ .

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Step 5.8.3.3.1.5.1

Factor $2$ out of $2 e^{5 y}$ .

$y' = - \frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}} - \frac{5 y^{2} e^{\frac{5 x}{2}}}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})} + \frac{2 (e^{5 y})}{- 10 x e^{5 y} + 4 y e^{\frac{5 x}{2}} + 4 y e^{x^{2} + y^{2}}}$

Step 5.8.3.3.1.5.2

Cancel the common factors.

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Step 5.8.3.3.1.5.2.1

Factor $2$ out of $- 10 x e^{5 y}$ .

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

Step 5.8.3.3.1.5.2.2

Factor $2$ out of $4 y e^{\frac{5 x}{2}}$ .

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

Step 5.8.3.3.1.5.2.3

Factor $2$ out of $2 (- 5 x e^{5 y}) + 2 (2 y e^{\frac{5 x}{2}})$ .

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

Step 5.8.3.3.1.5.2.4

Factor $2$ out of $4 y e^{x^{2} + y^{2}}$ .

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

Step 5.8.3.3.1.5.2.5

Factor $2$ out of $2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}}) + 2 (2 y e^{x^{2} + y^{2}})$ .

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

Step 5.8.3.3.1.5.2.6

Cancel the common factor.

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

Step 5.8.3.3.1.5.2.7

Rewrite the expression.

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

Step 5.8.3.3.2

To write $- \frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.8.3.3.3

Write each expression with a common denominator of $(- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.8.3.3.3.1

Multiply $\frac{2 x e^{x^{2} + y^{2}}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}}$ by $\frac{2}{2}$ .

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

Step 5.8.3.3.3.2

Reorder the factors of $(- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}) \cdot 2$ .

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

Step 5.8.3.3.4

Combine the numerators over the common denominator.

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

Step 5.8.3.3.5

Multiply $2$ by $- 2$ .

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

Step 5.8.3.3.6

To write $\frac{e^{5 y}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

Step 5.8.3.3.7

Write each expression with a common denominator of $2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.8.3.3.7.1

Multiply $\frac{e^{5 y}}{- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}}$ by $\frac{2}{2}$ .

$y' = \frac{- 4 x e^{x^{2} + y^{2}} - 5 y^{2} e^{\frac{5 x}{2}}}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})} + \frac{e^{5 y} \cdot 2}{(- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}) \cdot 2}$

Step 5.8.3.3.7.2

Reorder the factors of $(- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}}) \cdot 2$ .

$y' = \frac{- 4 x e^{x^{2} + y^{2}} - 5 y^{2} e^{\frac{5 x}{2}}}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})} + \frac{e^{5 y} \cdot 2}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})}$

Step 5.8.3.3.8

Combine the numerators over the common denominator.

$y' = \frac{- 4 x e^{x^{2} + y^{2}} - 5 y^{2} e^{\frac{5 x}{2}} + e^{5 y} \cdot 2}{2 (- 5 x e^{5 y} + 2 y e^{\frac{5 x}{2}} + 2 y e^{x^{2} + y^{2}})}$

Step 5.8.3.3.9

Move $2$ to the left of $e^{5 y}$ .

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

Step 5.8.3.3.10

Simplify terms.

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Step 5.8.3.3.10.1

Factor $- 1$ out of $- 4 x e^{x^{2} + y^{2}}$ .

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

Step 5.8.3.3.10.2

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

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

Step 5.8.3.3.10.3

Factor $- 1$ out of $- (4 x e^{x^{2} + y^{2}}) - (5 y^{2} e^{\frac{5 x}{2}})$ .

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

Step 5.8.3.3.10.4

Factor $- 1$ out of $2 e^{5 y}$ .

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

Step 5.8.3.3.10.5

Factor $- 1$ out of $- (4 x e^{x^{2} + y^{2}} + 5 y^{2} e^{\frac{5 x}{2}}) - (- 2 e^{5 y})$ .

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

Step 5.8.3.3.10.6

Rewrite $- (4 x e^{x^{2} + y^{2}} + 5 y^{2} e^{\frac{5 x}{2}} - 2 e^{5 y})$ as $- 1 (4 x e^{x^{2} + y^{2}} + 5 y^{2} e^{\frac{5 x}{2}} - 2 e^{5 y})$ .

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

Step 5.8.3.3.10.7

Factor $- 1$ out of $- 5 x e^{5 y}$ .

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

Step 5.8.3.3.10.8

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

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

Step 5.8.3.3.10.9

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

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

Step 5.8.3.3.10.10

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

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

Step 5.8.3.3.10.11

Factor $- 1$ out of $- (5 x e^{5 y} - 2 y e^{\frac{5 x}{2}}) - (- 2 y e^{x^{2} + y^{2}})$ .

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

Step 5.8.3.3.10.12

Rewrite $- (5 x e^{5 y} - 2 y e^{\frac{5 x}{2}} - 2 y e^{x^{2} + y^{2}})$ as $- 1 (5 x e^{5 y} - 2 y e^{\frac{5 x}{2}} - 2 y e^{x^{2} + y^{2}})$ .

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

Step 5.8.3.3.10.13

Cancel the common factor.

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

Step 5.8.3.3.10.14

Rewrite the expression.

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

Step 6

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

$\frac{d y}{d x} = \frac{4 x e^{x^{2} + y^{2}} + 5 y^{2} e^{\frac{5 x}{2}} - 2 e^{5 y}}{2 (5 x e^{5 y} - 2 y e^{\frac{5 x}{2}} - 2 y e^{x^{2} + y^{2}})}$