Calculus Examples

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

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = e^{x} (y^{3} + x y^{3} + 1)$ .

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [e^{x} (y^{3} + x y^{3} + 1)]$

Step 1.2

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

$\frac{\partial M}{\partial y} = e^{x} \frac{d}{d y} [y^{3} + x y^{3} + 1]$

Step 1.3

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

$\frac{\partial M}{\partial y} = e^{x} (\frac{d}{d y} [y^{3}] + \frac{d}{d y} [x y^{3}] + \frac{d}{d y} [1])$

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Move $3$ to the left of $x$ .

$\frac{\partial M}{\partial y} = e^{x} (3 y^{2} + 3 \cdot x y^{2} + \frac{d}{d y} [1])$

Step 1.8

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

$\frac{\partial M}{\partial y} = e^{x} (3 y^{2} + 3 x y^{2} + 0)$

Step 1.9

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

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

Step 1.10

Simplify.

Tap for more steps...

Step 1.10.1

Apply the distributive property.

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

Step 1.10.2

Move $3$ to the left of $e^{x}$ .

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

Step 1.10.3

Reorder terms.

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

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

Since $3 y^{2}$ is constant with respect to $x$ , the derivative of $3 y^{2} (x e^{x} - 6)$ with respect to $x$ is $3 y^{2} \frac{d}{d x} [x e^{x} - 6]$ .

$\frac{\partial N}{\partial x} = 3 y^{2} \frac{d}{d x} [x e^{x} - 6]$

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Differentiate.

Tap for more steps...

Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

$\frac{\partial N}{\partial x} = 3 y^{2} (x e^{x} + e^{x} + 0)$

Step 2.5.4

Add $x e^{x} + e^{x}$ and $0$ .

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

Step 2.6

Simplify.

Tap for more steps...

Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Reorder terms.

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Since the two sides have been shown to be equivalent, the equation is an identity.

$3 e^{x} y^{2} + 3 e^{x} y^{2} x = 3 e^{x} y^{2} x + 3 e^{x} y^{2}$ is an identity.

Step 4

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

$f (x, y) = \int 3 y^{2} (x e^{x} - 6) d y$

Step 5

Integrate $N (x, y) = 3 y^{2} (x e^{x} - 6)$ to find $f (x, y)$ .

Tap for more steps...

Step 5.1

Since $3 (x e^{x} - 6)$ is constant with respect to $y$ , move $3 (x e^{x} - 6)$ out of the integral.

$f (x, y) = 3 (x e^{x} - 6) \int y^{2} d y$

Step 5.2

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

$f (x, y) = 3 (x e^{x} - 6) (\frac{1}{3} y^{3} + C)$

Step 5.3

Simplify the answer.

Tap for more steps...

Step 5.3.1

Rewrite $3 (x e^{x} - 6) (\frac{1}{3} y^{3} + C)$ as $(3 x e^{x} - 18) (\frac{1}{3} y^{3}) + C$ .

$f (x, y) = (3 x e^{x} - 18) (\frac{1}{3} y^{3}) + C$

Step 5.3.2

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

$f (x, y) = (3 x e^{x} - 18) \frac{y^{3}}{3} + C$

Step 5.3.3

Simplify.

Tap for more steps...

Step 5.3.3.1

Reorder terms.

$f (x, y) = (3 x e^{x} - 18) (\frac{1}{3} y^{3}) + C$

Step 5.3.3.2

Remove parentheses.

$f (x, y) = (3 x e^{x} - 18) \frac{1}{3} y^{3} + C$

Step 6

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

$f (x, y) = (3 x e^{x} - 18) \frac{1}{3} y^{3} + g (x)$

Step 7

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

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

Step 8

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

Tap for more steps...

Step 8.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [(3 x e^{x} - 18) \frac{1}{3} y^{3} + g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.2

By the Sum Rule, the derivative of $(3 x e^{x} - 18) \frac{1}{3} y^{3} + g (x)$ with respect to $x$ is $\frac{d}{d x} [(3 x e^{x} - 18) \frac{1}{3} y^{3}] + \frac{d}{d x} [g (x)]$ .

$\frac{d}{d x} [(3 x e^{x} - 18) \frac{1}{3} y^{3}] + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3

Evaluate $\frac{d}{d x} [(3 x e^{x} - 18) \frac{1}{3} y^{3}]$ .

Tap for more steps...

Step 8.3.1

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

$\frac{d}{d x} [(3 x e^{x} - 18) \frac{y^{3}}{3}] + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3.2

Since $\frac{y^{3}}{3}$ is constant with respect to $x$ , the derivative of $(3 x e^{x} - 18) \frac{y^{3}}{3}$ with respect to $x$ is $\frac{y^{3}}{3} \frac{d}{d x} [3 x e^{x} - 18]$ .

$\frac{y^{3}}{3} \frac{d}{d x} [3 x e^{x} - 18] + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3.3

By the Sum Rule, the derivative of $3 x e^{x} - 18$ with respect to $x$ is $\frac{d}{d x} [3 x e^{x}] + \frac{d}{d x} [- 18]$ .

$\frac{y^{3}}{3} (\frac{d}{d x} [3 x e^{x}] + \frac{d}{d x} [- 18]) + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3.4

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

$\frac{y^{3}}{3} (3 \frac{d}{d x} [x e^{x}] + \frac{d}{d x} [- 18]) + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3.5

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

$\frac{y^{3}}{3} (3 (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x]) + \frac{d}{d x} [- 18]) + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3.6

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

$\frac{y^{3}}{3} (3 (x e^{x} + e^{x} \frac{d}{d x} [x]) + \frac{d}{d x} [- 18]) + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3.7

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

$\frac{y^{3}}{3} (3 (x e^{x} + e^{x} \cdot 1) + \frac{d}{d x} [- 18]) + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3.8

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18$ with respect to $x$ is $0$ .

$\frac{y^{3}}{3} (3 (x e^{x} + e^{x} \cdot 1) + 0) + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3.9

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

$\frac{y^{3}}{3} (3 (x e^{x} + e^{x}) + 0) + \frac{d}{d x} [g (x)] = e^{x} (y^{3} + x y^{3} + 1)$

Step 8.3.10

Add $3 (x e^{x} + e^{x})$ and $0$ .

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

Step 8.3.11

Combine $3$ and $\frac{y^{3}}{3}$ .

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

Step 8.3.12

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.3.12.1

Cancel the common factor.

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

Step 8.3.12.2

Divide $y^{3}$ by $1$ .

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

Step 8.4

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

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

Step 8.5

Simplify.

Tap for more steps...

Step 8.5.1

Apply the distributive property.

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

Step 8.5.2

Reorder terms.

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

Step 9

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

Tap for more steps...

Step 9.1

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

Tap for more steps...

Step 9.1.1

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

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

Step 9.1.2

Simplify $e^{x} (y^{3} + x y^{3} + 1)$ .

Tap for more steps...

Step 9.1.2.1

Apply the distributive property.

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

Step 9.1.2.2

Simplify the expression.

Tap for more steps...

Step 9.1.2.2.1

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

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

Step 9.1.2.2.2

Reorder factors in $e^{x} y^{3} + e^{x} (x y^{3}) + e^{x}$ .

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

Step 9.1.3

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

Tap for more steps...

Step 9.1.3.1

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

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

Step 9.1.3.2

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

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

Step 9.1.3.3

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

Tap for more steps...

Step 9.1.3.3.1

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

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

Step 9.1.3.3.2

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

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

Step 9.1.3.3.3

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

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

Step 9.1.3.3.4

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

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

Step 9.1.3.3.5

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

$\frac{d g}{d x} = e^{x}$

Step 10

Find the antiderivative of $e^{x}$ to find $g (x)$ .

Tap for more steps...

Step 10.1

Integrate both sides of $\frac{d g}{d x} = e^{x}$ .

$\int \frac{d g}{d x} d x = \int e^{x} d x$

Step 10.2

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

$g (x) = \int e^{x} d x$

Step 10.3

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$g (x) = e^{x} + C$

Step 11

Substitute for $g (x)$ in $f (x, y) = (3 x e^{x} - 18) \frac{1}{3} y^{3} + g (x)$ .

$f (x, y) = (3 x e^{x} - 18) \frac{1}{3} y^{3} + e^{x} + C$

Step 12

Simplify $f (x, y) = (3 x e^{x} - 18) \frac{1}{3} y^{3} + e^{x} + C$ .

Tap for more steps...

Step 12.1

Simplify each term.

Tap for more steps...

Step 12.1.1

Apply the distributive property.

$f (x, y) = (3 x e^{x} \frac{1}{3} - 18 (\frac{1}{3})) y^{3} + e^{x} + C$

Step 12.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 12.1.2.1

Factor $3$ out of $3 x e^{x}$ .

$f (x, y) = (3 (x e^{x}) \frac{1}{3} - 18 (\frac{1}{3})) y^{3} + e^{x} + C$

Step 12.1.2.2

Cancel the common factor.

$f (x, y) = (3 (x e^{x}) \frac{1}{3} - 18 (\frac{1}{3})) y^{3} + e^{x} + C$

Step 12.1.2.3

Rewrite the expression.

$f (x, y) = (x e^{x} - 18 (\frac{1}{3})) y^{3} + e^{x} + C$

Step 12.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 12.1.3.1

Factor $3$ out of $- 18$ .

$f (x, y) = (x e^{x} + 3 (- 6) \frac{1}{3}) y^{3} + e^{x} + C$

Step 12.1.3.2

Cancel the common factor.

$f (x, y) = (x e^{x} + 3 \cdot - 6 \frac{1}{3}) y^{3} + e^{x} + C$

Step 12.1.3.3

Rewrite the expression.

$f (x, y) = (x e^{x} - 6) y^{3} + e^{x} + C$

Step 12.1.4

Apply the distributive property.

$f (x, y) = x e^{x} y^{3} - 6 y^{3} + e^{x} + C$

Step 12.2

Reorder factors in $f (x, y) = x e^{x} y^{3} - 6 y^{3} + e^{x} + C$ .

$f (x, y) = x y^{3} e^{x} - 6 y^{3} + e^{x} + C$