Calculus Examples

$e^{x^{2}} y - 3 \sqrt{y^{2} + 2} = x^{2} + 1$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

Differentiate the left side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $e^{x^{2}} y - 3 {(y^{2} + 2)}^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [e^{x^{2}} y] + \frac{d}{d x} [- 3 {(y^{2} + 2)}^{\frac{1}{2}}]$ .

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

Step 3.2

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

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.4

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

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

Step 3.3

Evaluate $\frac{d}{d x} [- 3 {(y^{2} + 2)}^{\frac{1}{2}}]$ .

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Step 3.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 {(y^{2} + 2)}^{\frac{1}{2}}$ with respect to $x$ is $- 3 \frac{d}{d x} [{(y^{2} + 2)}^{\frac{1}{2}}]$ .

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 3.3.2.3

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

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

Step 3.3.3

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

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

Step 3.3.4

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 3.3.4.1

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

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

Step 3.3.4.2

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

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

Step 3.3.4.3

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

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

Step 3.3.5

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

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

Step 3.3.6

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$e^{x^{2}} y' + y e^{x^{2}} (2 x) - 3 (\frac{1}{2} {(y^{2} + 2)}^{\frac{1}{2} - 1} (2 y y' + 0))$

Step 3.3.7

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

$e^{x^{2}} y' + y e^{x^{2}} (2 x) - 3 (\frac{1}{2} {(y^{2} + 2)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (2 y y' + 0))$

Step 3.3.8

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

$e^{x^{2}} y' + y e^{x^{2}} (2 x) - 3 (\frac{1}{2} {(y^{2} + 2)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (2 y y' + 0))$

Step 3.3.9

Combine the numerators over the common denominator.

$e^{x^{2}} y' + y e^{x^{2}} (2 x) - 3 (\frac{1}{2} {(y^{2} + 2)}^{\frac{1 - 1 \cdot 2}{2}} (2 y y' + 0))$

Step 3.3.10

Simplify the numerator.

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Step 3.3.10.1

Multiply $- 1$ by $2$ .

$e^{x^{2}} y' + y e^{x^{2}} (2 x) - 3 (\frac{1}{2} {(y^{2} + 2)}^{\frac{1 - 2}{2}} (2 y y' + 0))$

Step 3.3.10.2

Subtract $2$ from $1$ .

$e^{x^{2}} y' + y e^{x^{2}} (2 x) - 3 (\frac{1}{2} {(y^{2} + 2)}^{\frac{- 1}{2}} (2 y y' + 0))$

Step 3.3.11

Move the negative in front of the fraction.

$e^{x^{2}} y' + y e^{x^{2}} (2 x) - 3 (\frac{1}{2} {(y^{2} + 2)}^{- \frac{1}{2}} (2 y y' + 0))$

Step 3.3.12

Add $2 y y'$ and $0$ .

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

Step 3.3.13

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

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

Step 3.3.14

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

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

Step 3.3.15

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

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

Step 3.3.16

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

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

Step 3.3.17

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

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

Step 3.3.18

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

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

Step 3.3.19

Cancel the common factor.

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

Step 3.3.20

Rewrite the expression.

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

Step 3.3.21

Combine $- 3$ and $\frac{y y'}{{(y^{2} + 2)}^{\frac{1}{2}}}$ .

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

Step 3.3.22

Move the negative in front of the fraction.

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

Step 3.4

Simplify.

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Step 3.4.1

Reorder terms.

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

Step 3.4.2

Reorder factors in $e^{x^{2}} y' + 2 e^{x^{2}} x y - \frac{3 y y'}{{(y^{2} + 2)}^{\frac{1}{2}}}$ .

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

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$2 x + 0$

Step 4.4

Add $2 x$ and $0$ .

$2 x$

Step 5

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

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

Step 6

Solve for $y'$ .

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Step 6.1

Find the LCD of the terms in the equation.

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Step 6.1.1

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

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

Step 6.1.2

The LCM of one and any expression is the expression.

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

Step 6.2

Multiply each term in $e^{x^{2}} y' + 2 x y e^{x^{2}} - \frac{3 y y'}{{(y^{2} + 2)}^{\frac{1}{2}}} = 2 x$ by ${(y^{2} + 2)}^{\frac{1}{2}}$ to eliminate the fractions.

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

Move the leading negative in $- \frac{3 y y'}{{(y^{2} + 2)}^{\frac{1}{2}}}$ into the numerator.

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

Step 6.2.2.1.2

Cancel the common factor.

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

Step 6.2.2.1.3

Rewrite the expression.

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

Step 6.2.2.2

Reorder factors in $e^{x^{2}} y' {(y^{2} + 2)}^{\frac{1}{2}} + 2 x y e^{x^{2}} {(y^{2} + 2)}^{\frac{1}{2}} - 3 y y'$ .

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

Step 6.3

Solve the equation.

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Step 6.3.1

Subtract $2 x y e^{x^{2}} {(y^{2} + 2)}^{\frac{1}{2}}$ from both sides of the equation.

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

Step 6.3.2

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

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Step 6.3.2.1

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

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

Step 6.3.2.2

Factor $y'$ out of $- 3 y y'$ .

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

Step 6.3.2.3

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

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

Step 6.3.3

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

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Step 6.3.3.1

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

Step 6.3.3.2

Simplify the left side.

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Step 6.3.3.2.1

Cancel the common factor.

Step 6.3.3.2.2

Divide $y'$ by $1$ .

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

Step 6.3.3.3

Simplify the right side.

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Step 6.3.3.3.1

Combine the numerators over the common denominator.

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

Step 6.3.3.3.2

Simplify the numerator.

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Step 6.3.3.3.2.1

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

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Step 6.3.3.3.2.1.1

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

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

Step 6.3.3.3.2.1.2

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

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

Step 6.3.3.3.2.1.3

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

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

Step 6.3.3.3.2.2

Rewrite $- 1 y$ as $- y$ .

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

Step 7

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

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