Calculus Examples

$3 (x^{2} + y^{2}) d x + x (x^{2} + 3 y^{2} + 6 y) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 3 (x^{2} + y^{2})$ .

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [3 (x^{2} + y^{2})]$

Step 1.2

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

$\frac{\partial M}{\partial y} = 3 \frac{d}{d y} [x^{2} + y^{2}]$

Step 1.3

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

$\frac{\partial M}{\partial y} = 3 (\frac{d}{d y} [x^{2}] + \frac{d}{d y} [y^{2}])$

Step 1.4

Since $x^{2}$ is constant with respect to $y$ , the derivative of $x^{2}$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 3 (0 + \frac{d}{d y} [y^{2}])$

Step 1.5

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

$\frac{\partial M}{\partial y} = 3 \frac{d}{d y} [y^{2}]$

Step 1.6

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

$\frac{\partial M}{\partial y} = 3 (2 y)$

Step 1.7

Multiply $2$ by $3$ .

$\frac{\partial M}{\partial y} = 6 y$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = x (x^{2} + 3 y^{2} + 6 y)$ .

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x (x^{2} + 3 y^{2} + 6 y)]$

Step 2.2

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 $f (x) = x$ and $g (x) = x^{2} + 3 y^{2} + 6 y$ .

$\frac{\partial N}{\partial x} = x \frac{d}{d x} [x^{2} + 3 y^{2} + 6 y] + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

By the Sum Rule, the derivative of $x^{2} + 3 y^{2} + 6 y$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 y]$ .

$\frac{\partial N}{\partial x} = x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 y]) + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.3.2

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

$\frac{\partial N}{\partial x} = x (2 x + \frac{d}{d x} [3 y^{2}] + \frac{d}{d x} [6 y]) + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.3.3

Since $3 y^{2}$ is constant with respect to $x$ , the derivative of $3 y^{2}$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = x (2 x + 0 + \frac{d}{d x} [6 y]) + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.3.4

Add $2 x$ and $0$ .

$\frac{\partial N}{\partial x} = x (2 x + \frac{d}{d x} [6 y]) + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.3.5

Since $6 y$ is constant with respect to $x$ , the derivative of $6 y$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = x (2 x + 0) + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.3.6

Add $2 x$ and $0$ .

$\frac{\partial N}{\partial x} = x (2 x) + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.4

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

$\frac{\partial N}{\partial x} = 2 (x^{1} x) + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.5

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

$\frac{\partial N}{\partial x} = 2 (x^{1} x^{1}) + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.6

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

$\frac{\partial N}{\partial x} = 2 x^{1 + 1} + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.7

Add $1$ and $1$ .

$\frac{\partial N}{\partial x} = 2 x^{2} + (x^{2} + 3 y^{2} + 6 y) \frac{d}{d x} [x]$

Step 2.8

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

$\frac{\partial N}{\partial x} = 2 x^{2} + (x^{2} + 3 y^{2} + 6 y) \cdot 1$

Step 2.9

Simplify by adding terms.

Tap for more steps...

Step 2.9.1

Multiply $x^{2} + 3 y^{2} + 6 y$ by $1$ .

$\frac{\partial N}{\partial x} = 2 x^{2} + x^{2} + 3 y^{2} + 6 y$

Step 2.9.2

Add $2 x^{2}$ and $x^{2}$ .

$\frac{\partial N}{\partial x} = 3 x^{2} + 3 y^{2} + 6 y$

Step 3

Check that $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

Tap for more steps...

Step 3.1

Substitute $6 y$ for $\frac{\partial M}{\partial y}$ and $3 x^{2} + 3 y^{2} + 6 y$ for $\frac{\partial N}{\partial x}$ .

$6 y = 3 x^{2} + 3 y^{2} + 6 y$

Step 3.2

Since the left side does not equal the right side, the equation is not an identity.

$6 y = 3 x^{2} + 3 y^{2} + 6 y$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

Tap for more steps...

Step 4.1

Substitute $3 x^{2} + 3 y^{2} + 6 y$ for $\frac{\partial N}{\partial x}$ .

$\frac{3 x^{2} + 3 y^{2} + 6 y - \frac{\partial M}{\partial y}}{M}$

Step 4.2

Substitute $6 y$ for $\frac{\partial M}{\partial y}$ .

$\frac{3 x^{2} + 3 y^{2} + 6 y - (6 y)}{M}$

Step 4.3

Substitute $3 (x^{2} + y^{2})$ for $M$ .

Tap for more steps...

Step 4.3.1

Substitute $3 (x^{2} + y^{2})$ for $M$ .

$\frac{3 x^{2} + 3 y^{2} + 6 y - (6 y)}{3 (x^{2} + y^{2})}$

Step 4.3.2

Cancel the common factor of $3 x^{2} + 3 y^{2} + 6 y - (6 y)$ and $3$ .

Tap for more steps...

Step 4.3.2.1

Factor $- 1$ out of $3 x^{2}$ .

$\frac{- (- 3 x^{2}) + 3 y^{2} + 6 y - (6 y)}{3 (x^{2} + y^{2})}$

Step 4.3.2.2

Factor $- 1$ out of $3 y^{2}$ .

$\frac{- (- 3 x^{2}) - (- 3 y^{2}) + 6 y - (6 y)}{3 (x^{2} + y^{2})}$

Step 4.3.2.3

Factor $- 1$ out of $- (- 3 x^{2}) - (- 3 y^{2})$ .

$\frac{- (- 3 x^{2} - 3 y^{2}) + 6 y - (6 y)}{3 (x^{2} + y^{2})}$

Step 4.3.2.4

Factor $- 1$ out of $6 y$ .

$\frac{- (- 3 x^{2} - 3 y^{2}) - (- 6 y) - (6 y)}{3 (x^{2} + y^{2})}$

Step 4.3.2.5

Factor $- 1$ out of $- (- 3 x^{2} - 3 y^{2}) - (- 6 y)$ .

$\frac{- (- 3 x^{2} - 3 y^{2} - 6 y) - (6 y)}{3 (x^{2} + y^{2})}$

Step 4.3.2.6

Factor $- 1$ out of $- (- 3 x^{2} - 3 y^{2} - 6 y) - (6 y)$ .

$\frac{- (- 3 x^{2} - 3 y^{2} - 6 y + 6 y)}{3 (x^{2} + y^{2})}$

Step 4.3.2.7

Rewrite $- (- 3 x^{2} - 3 y^{2} - 6 y + 6 y)$ as $- 1 (- 3 x^{2} - 3 y^{2} - 6 y + 6 y)$ .

$\frac{- 1 (- 3 x^{2} - 3 y^{2} - 6 y + 6 y)}{3 (x^{2} + y^{2})}$

Step 4.3.2.8

Factor $3$ out of $- 1 (- 3 x^{2} - 3 y^{2} - 6 y + 6 y)$ .

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

Step 4.3.2.9

Cancel the common factors.

Tap for more steps...

Step 4.3.2.9.1

Cancel the common factor.

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

Step 4.3.2.9.2

Rewrite the expression.

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

Step 4.3.3

Simplify the numerator.

Tap for more steps...

Step 4.3.3.1

Add $- 2 y$ and $2 y$ .

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

Step 4.3.3.2

Add $- x^{2} - y^{2}$ and $0$ .

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

Step 4.3.4

Cancel the common factor of $- x^{2} - y^{2}$ and $x^{2} + y^{2}$ .

Tap for more steps...

Step 4.3.4.1

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

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

Step 4.3.4.2

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

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

Step 4.3.4.3

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

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

Step 4.3.4.4

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

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

Step 4.3.4.5

Cancel the common factor.

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

Step 4.3.4.6

Divide $- 1 \cdot (- 1)$ by $1$ .

$- 1 \cdot (- 1)$

Step 4.3.5

Rewrite $- 1 \cdot (- 1)$ as $- (- 1)$ .

$- (- 1)$

Step 4.3.6

Substitute $3 (x^{2} + y^{2})$ for $M$ .

$1$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

$μ (x, y) = e^{\int d y}$

Step 5

Evaluate the integral $e^{\int d y}$ .

Tap for more steps...

Step 5.1

Apply the constant rule.

$μ (x, y) = e^{y + C}$

Step 5.2

Simplify.

$μ (x, y) = e^{y} + C$

Step 6

Multiply both sides of $3 (x^{2} + y^{2}) d x + x (x^{2} + 3 y^{2} + 6 y) d y = 0$ by the integration factor $e^{y}$ .

Tap for more steps...

Step 6.1

Multiply $3 (x^{2} + y^{2})$ by $e^{y}$ .

$3 (x^{2} + y^{2}) e^{y} d x + x (x^{2} + 3 y^{2} + 6 y) d y = 0$

Step 6.2

Apply the distributive property.

$(3 x^{2} + 3 y^{2}) e^{y} d x + x (x^{2} + 3 y^{2} + 6 y) d y = 0$

Step 6.3

Apply the distributive property.

$(3 x^{2} e^{y} + 3 y^{2} e^{y}) d x + x (x^{2} + 3 y^{2} + 6 y) d y = 0$

Step 6.4

Multiply $x (x^{2} + 3 y^{2} + 6 y)$ by $e^{y}$ .

$(3 x^{2} e^{y} + 3 y^{2} e^{y}) d x + x (x^{2} + 3 y^{2} + 6 y) e^{y} d y = 0$

Step 6.5

Apply the distributive property.

$(3 x^{2} e^{y} + 3 y^{2} e^{y}) d x + (x \cdot x^{2} + x (3 y^{2}) + x (6 y)) e^{y} d y = 0$

Step 6.6

Simplify.

Tap for more steps...

Step 6.6.1

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

Tap for more steps...

Step 6.6.1.1

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

Tap for more steps...

Step 6.6.1.1.1

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

$(3 x^{2} e^{y} + 3 y^{2} e^{y}) d x + (x^{1} x^{2} + x (3 y^{2}) + x (6 y)) e^{y} d y = 0$

Step 6.6.1.1.2

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

$(3 x^{2} e^{y} + 3 y^{2} e^{y}) d x + (x^{1 + 2} + x (3 y^{2}) + x (6 y)) e^{y} d y = 0$

Step 6.6.1.2

Add $1$ and $2$ .

$(3 x^{2} e^{y} + 3 y^{2} e^{y}) d x + (x^{3} + x (3 y^{2}) + x (6 y)) e^{y} d y = 0$

Step 6.6.2

Rewrite using the commutative property of multiplication.

$(3 x^{2} e^{y} + 3 y^{2} e^{y}) d x + (x^{3} + 3 x y^{2} + x (6 y)) e^{y} d y = 0$

Step 6.6.3

Rewrite using the commutative property of multiplication.

$(3 x^{2} e^{y} + 3 y^{2} e^{y}) d x + (x^{3} + 3 x y^{2} + 6 x y) e^{y} d y = 0$

Step 6.7

Apply the distributive property.

$(3 x^{2} e^{y} + 3 y^{2} e^{y}) d x + (x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}) d y = 0$

Step 7

Set $f (x, y)$ equal to the integral of $M (x, y)$ .

$f (x, y) = \int 3 x^{2} e^{y} + 3 y^{2} e^{y} d x$

Step 8

Integrate $M (x, y) = 3 x^{2} e^{y} + 3 y^{2} e^{y}$ to find $f (x, y)$ .

Tap for more steps...

Step 8.1

Split the single integral into multiple integrals.

$f (x, y) = \int 3 x^{2} e^{y} d x + \int 3 y^{2} e^{y} d x$

Step 8.2

Since $3 e^{y}$ is constant with respect to $x$ , move $3 e^{y}$ out of the integral.

$f (x, y) = 3 e^{y} \int x^{2} d x + \int 3 y^{2} e^{y} d x$

Step 8.3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$f (x, y) = 3 e^{y} (\frac{1}{3} x^{3} + C) + \int 3 y^{2} e^{y} d x$

Step 8.4

Apply the constant rule.

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

Step 8.5

Combine $\frac{1}{3}$ and $x^{3}$ .

$f (x, y) = 3 e^{y} (\frac{x^{3}}{3} + C) + 3 y^{2} e^{y} x + C$

Step 8.6

Simplify.

$f (x, y) = e^{y} x^{3} + 3 y^{2} e^{y} x + C$

Step 9

Since the integral of $g (y)$ will contain an integration constant, we can replace $C$ with $g (y)$ .

$f (x, y) = e^{y} x^{3} + 3 y^{2} e^{y} x + g (y)$

Step 10

Set $\frac{\partial f}{\partial y} = N (x, y)$ .

$\frac{\partial f}{\partial y} = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11

Find $\frac{\partial f}{\partial y}$ .

Tap for more steps...

Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [e^{y} x^{3} + 3 y^{2} e^{y} x + g (y)] = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.2

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

$\frac{d}{d y} [e^{y} x^{3}] + \frac{d}{d y} [3 y^{2} e^{y} x] + \frac{d}{d y} [g (y)] = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.3

Evaluate $\frac{d}{d y} [e^{y} x^{3}]$ .

Tap for more steps...

Step 11.3.1

Since $x^{3}$ is constant with respect to $y$ , the derivative of $e^{y} x^{3}$ with respect to $y$ is $x^{3} \frac{d}{d y} [e^{y}]$ .

$x^{3} \frac{d}{d y} [e^{y}] + \frac{d}{d y} [3 y^{2} e^{y} x] + \frac{d}{d y} [g (y)] = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.3.2

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

$x^{3} e^{y} + \frac{d}{d y} [3 y^{2} e^{y} x] + \frac{d}{d y} [g (y)] = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.4

Evaluate $\frac{d}{d y} [3 y^{2} e^{y} x]$ .

Tap for more steps...

Step 11.4.1

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

$x^{3} e^{y} + 3 x \frac{d}{d y} [y^{2} e^{y}] + \frac{d}{d y} [g (y)] = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.4.2

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

$x^{3} e^{y} + 3 x (y^{2} \frac{d}{d y} [e^{y}] + e^{y} \frac{d}{d y} [y^{2}]) + \frac{d}{d y} [g (y)] = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.4.3

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

$x^{3} e^{y} + 3 x (y^{2} e^{y} + e^{y} \frac{d}{d y} [y^{2}]) + \frac{d}{d y} [g (y)] = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.4.4

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

$x^{3} e^{y} + 3 x (y^{2} e^{y} + e^{y} (2 y)) + \frac{d}{d y} [g (y)] = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.5

Differentiate using the function rule which states that the derivative of $g (y)$ is $\frac{d g}{d y}$ .

$x^{3} e^{y} + 3 x (y^{2} e^{y} + e^{y} (2 y)) + \frac{d g}{d y} = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.6

Simplify.

Tap for more steps...

Step 11.6.1

Apply the distributive property.

$x^{3} e^{y} + 3 x (y^{2} e^{y}) + 3 x (e^{y} (2 y)) + \frac{d g}{d y} = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.6.2

Multiply $2$ by $3$ .

$x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x e^{y} y + \frac{d g}{d y} = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.6.3

Reorder terms.

$\frac{d g}{d y} + e^{y} x^{3} + 3 e^{y} y^{2} x + 6 e^{y} x y = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 11.6.4

Reorder factors in $\frac{d g}{d y} + e^{y} x^{3} + 3 e^{y} y^{2} x + 6 e^{y} x y$ .

$\frac{d g}{d y} + x^{3} e^{y} + 3 y^{2} x e^{y} + 6 x y e^{y} = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y}$

Step 12

Solve for $\frac{d g}{d y}$ .

Tap for more steps...

Step 12.1

Move all terms not containing $\frac{d g}{d y}$ to the right side of the equation.

Tap for more steps...

Step 12.1.1

Subtract $x^{3} e^{y}$ from both sides of the equation.

$\frac{d g}{d y} + 3 y^{2} x e^{y} + 6 x y e^{y} = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y} - x^{3} e^{y}$

Step 12.1.2

Subtract $3 y^{2} x e^{y}$ from both sides of the equation.

$\frac{d g}{d y} + 6 x y e^{y} = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y} - x^{3} e^{y} - 3 y^{2} x e^{y}$

Step 12.1.3

Subtract $6 x y e^{y}$ from both sides of the equation.

$\frac{d g}{d y} = x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y} - x^{3} e^{y} - 3 y^{2} x e^{y} - 6 x y e^{y}$

Step 12.1.4

Combine the opposite terms in $x^{3} e^{y} + 3 x y^{2} e^{y} + 6 x y e^{y} - x^{3} e^{y} - 3 y^{2} x e^{y} - 6 x y e^{y}$ .

Tap for more steps...

Step 12.1.4.1

Subtract $x^{3} e^{y}$ from $x^{3} e^{y}$ .

$\frac{d g}{d y} = 3 x y^{2} e^{y} + 6 x y e^{y} + 0 - 3 y^{2} x e^{y} - 6 x y e^{y}$

Step 12.1.4.2

Add $3 x y^{2} e^{y} + 6 x y e^{y}$ and $0$ .

$\frac{d g}{d y} = 3 x y^{2} e^{y} + 6 x y e^{y} - 3 y^{2} x e^{y} - 6 x y e^{y}$

Step 12.1.4.3

Reorder the factors in the terms $3 x y^{2} e^{y}$ and $- 3 y^{2} x e^{y}$ .

$\frac{d g}{d y} = 3 e^{y} y^{2} x + 6 x y e^{y} - 3 e^{y} y^{2} x - 6 x y e^{y}$

Step 12.1.4.4

Subtract $3 e^{y} y^{2} x$ from $3 e^{y} y^{2} x$ .

$\frac{d g}{d y} = 6 x y e^{y} + 0 - 6 x y e^{y}$

Step 12.1.4.5

Add $6 x y e^{y}$ and $0$ .

$\frac{d g}{d y} = 6 x y e^{y} - 6 x y e^{y}$

Step 12.1.4.6

Subtract $6 x y e^{y}$ from $6 x y e^{y}$ .

$\frac{d g}{d y} = 0$

Step 13

Find the antiderivative of $0$ to find $g (y)$ .

Tap for more steps...

Step 13.1

Integrate both sides of $\frac{d g}{d y} = 0$ .

$\int \frac{d g}{d y} d y = \int 0 d y$

Step 13.2

Evaluate $\int \frac{d g}{d y} d y$ .

$g (y) = \int 0 d y$

Step 13.3

The integral of $0$ with respect to $y$ is $0$ .

$g (y) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (y) = C$

Step 14

Substitute for $g (y)$ in $f (x, y) = e^{y} x^{3} + 3 y^{2} e^{y} x + g (y)$ .

$f (x, y) = e^{y} x^{3} + 3 y^{2} e^{y} x + C$

Step 15

Reorder factors in $f (x, y) = e^{y} x^{3} + 3 y^{2} e^{y} x + C$ .

$f (x, y) = x^{3} e^{y} + 3 y^{2} x e^{y} + C$