Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Multiply $3$ by $1$ .

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Simplify.

Tap for more steps...

Step 1.9.1

Reorder the factors of $e^{x^{3}} (3 x^{2})$ .

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

Step 1.9.2

Reorder factors in $3 e^{x^{3}} x^{2}$ .

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

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [e^{x^{3}}]$

Step 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 $g (x) = x^{3}$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$\frac{\partial N}{\partial x} = \frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{3}]$

Step 2.2.2

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

$\frac{\partial N}{\partial x} = e^{u} \frac{d}{d x} [x^{3}]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{3}$ .

$\frac{\partial N}{\partial x} = e^{x^{3}} \frac{d}{d x} [x^{3}]$

Step 2.3

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

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Reorder the factors of $e^{x^{3}} (3 x^{2})$ .

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

Step 2.4.2

Reorder factors in $3 e^{x^{3}} x^{2}$ .

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

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

Tap for more steps...

Step 5.1

Apply the constant rule.

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

Step 6

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

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

Step 7

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

$\frac{\partial f}{\partial x} = e^{x^{3}} (3 x^{2} y - x^{2})$

Step 8

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

Tap for more steps...

Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

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

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

Step 8.3

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

Tap for more steps...

Step 8.3.1

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

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

Step 8.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{3}$ .

Tap for more steps...

Step 8.3.2.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

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

Step 8.3.2.2

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

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

Step 8.3.2.3

Replace all occurrences of $u$ with $x^{3}$ .

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

Step 8.3.3

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

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

Step 8.4

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

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

Step 8.5

Simplify.

Tap for more steps...

Step 8.5.1

Reorder terms.

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

Step 8.5.2

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

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

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

Simplify $e^{x^{3}} (3 x^{2} y - x^{2})$ .

Tap for more steps...

Step 9.1.1.1

Rewrite.

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

Step 9.1.1.2

Simplify by adding zeros.

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

Step 9.1.1.3

Apply the distributive property.

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

Step 9.1.1.4

Reorder.

Tap for more steps...

Step 9.1.1.4.1

Rewrite using the commutative property of multiplication.

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

Step 9.1.1.4.2

Rewrite using the commutative property of multiplication.

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

Step 9.1.1.4.3

Reorder factors in $3 e^{x^{3}} (x^{2} y) - e^{x^{3}} x^{2}$ .

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

Step 9.1.2

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

Tap for more steps...

Step 9.1.2.1

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

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

Step 9.1.2.2

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

Tap for more steps...

Step 9.1.2.2.1

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

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

Step 9.1.2.2.2

Add $- x^{2} e^{x^{3}}$ and $0$ .

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

Step 10

Find the antiderivative of $- x^{2} e^{x^{3}}$ to find $g (x)$ .

Tap for more steps...

Step 10.1

Integrate both sides of $\frac{d g}{d x} = - x^{2} e^{x^{3}}$ .

$\int \frac{d g}{d x} d x = \int - x^{2} e^{x^{3}} d x$

Step 10.2

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

$g (x) = \int - x^{2} e^{x^{3}} d x$

Step 10.3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$g (x) = - \int x^{2} e^{x^{3}} d x$

Step 10.4

Let $u_{2} = e^{x^{3}}$ . Then $d u_{2} = 3 x^{2} e^{x^{3}} d x$ , so $\frac{1}{3} d u_{2} = x^{2} e^{x^{3}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 10.4.1

Let $u_{2} = e^{x^{3}}$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 10.4.1.1

Differentiate $e^{x^{3}}$ .

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

Step 10.4.1.2

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 $g (x) = x^{3}$ .

Tap for more steps...

Step 10.4.1.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [x^{3}]$

Step 10.4.1.2.2

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

$e^{u_{1}} \frac{d}{d x} [x^{3}]$

Step 10.4.1.2.3

Replace all occurrences of $u_{1}$ with $x^{3}$ .

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

Step 10.4.1.3

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

$e^{x^{3}} (3 x^{2})$

Step 10.4.1.4

Simplify.

Tap for more steps...

Step 10.4.1.4.1

Reorder the factors of $e^{x^{3}} (3 x^{2})$ .

$3 e^{x^{3}} x^{2}$

Step 10.4.1.4.2

Reorder factors in $3 e^{x^{3}} x^{2}$ .

$3 x^{2} e^{x^{3}}$

Step 10.4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$g (x) = - \int \frac{1}{3} d u_{2}$

Step 10.5

Apply the constant rule.

$g (x) = - (\frac{1}{3} u_{2} + C)$

Step 10.6

Simplify the answer.

Tap for more steps...

Step 10.6.1

Rewrite $- (\frac{1}{3} u_{2} + C)$ as $- \frac{1}{3} u_{2} + C$ .

$g (x) = - \frac{1}{3} u_{2} + C$

Step 10.6.2

Replace all occurrences of $u_{2}$ with $e^{x^{3}}$ .

$g (x) = - \frac{1}{3} e^{x^{3}} + C$

Step 11

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

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

Step 12

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

Tap for more steps...

Step 12.1

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

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

Step 12.2

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

Tap for more steps...

Step 12.2.1

Reorder $e^{x^{3}}$ and $y$ .

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

Step 12.2.2

To write $y e^{x^{3}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 12.2.3

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

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

Step 12.2.4

Combine the numerators over the common denominator.

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

Step 12.3

Simplify the numerator.

Tap for more steps...

Step 12.3.1

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

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

Step 12.3.2

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

Tap for more steps...

Step 12.3.2.1

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

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

Step 12.3.2.2

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

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

Step 12.3.2.3

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

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