Calculus Examples

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

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = \frac{1}{x^{2}} (1 - x^{2} y)$ .

Tap for more steps...

Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

$\frac{\partial M}{\partial y} = \frac{1}{x^{2}} \frac{d}{d y} [1 - x^{2} y]$

Step 1.3

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

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

Step 1.4

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

$\frac{\partial M}{\partial y} = \frac{1}{x^{2}} (0 + \frac{d}{d y} [- x^{2} y])$

Step 1.5

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

$\frac{\partial M}{\partial y} = \frac{1}{x^{2}} \frac{d}{d y} [- x^{2} y]$

Step 1.6

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

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

Step 1.7

Simplify terms.

Tap for more steps...

Step 1.7.1

Combine $\frac{1}{x^{2}}$ and $x^{2}$ .

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

Step 1.7.2

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 1.7.2.1

Cancel the common factor.

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

Step 1.7.2.2

Rewrite the expression.

$\frac{\partial M}{\partial y} = - 1 \cdot 1 \frac{d}{d y} [y]$

Step 1.7.3

Multiply $- 1$ by $1$ .

$\frac{\partial M}{\partial y} = - \frac{d}{d y} [y]$

Step 1.8

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

$\frac{\partial M}{\partial y} = - 1 \cdot 1$

Step 1.9

Multiply $- 1$ by $1$ .

$\frac{\partial M}{\partial y} = - 1$

Step 2

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

Tap for more steps...

Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [y - x]$

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

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

$\frac{\partial N}{\partial x} = \frac{d}{d x} [y] + \frac{d}{d x} [- x]$

Step 2.2.2

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

$\frac{\partial N}{\partial x} = 0 + \frac{d}{d x} [- x]$

Step 2.3

Evaluate $\frac{d}{d x} [- x]$ .

Tap for more steps...

Step 2.3.1

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

$\frac{\partial N}{\partial x} = 0 - \frac{d}{d x} [x]$

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 0 - 1 \cdot 1$

Step 2.3.3

Multiply $- 1$ by $1$ .

$\frac{\partial N}{\partial x} = 0 - 1$

Step 2.4

Subtract $1$ from $0$ .

$\frac{\partial N}{\partial x} = - 1$

Step 3

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

Tap for more steps...

Step 3.1

Substitute $- 1$ for $\frac{\partial M}{\partial y}$ and $- 1$ for $\frac{\partial N}{\partial x}$ .

$- 1 = - 1$

Step 3.2

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

$- 1 = - 1$ is an identity.

Step 4

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

$f (x, y) = \int y - x d y$

Step 5

Integrate $N (x, y) = y - x$ to find $f (x, y)$ .

Tap for more steps...

Step 5.1

Split the single integral into multiple integrals.

$f (x, y) = \int y d y + \int - x d y$

Step 5.2

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

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

Step 5.3

Apply the constant rule.

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

Step 5.4

Simplify.

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

Step 6

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

$f (x, y) = \frac{1}{2} y^{2} - x y + g (x)$

Step 7

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

$\frac{\partial f}{\partial x} = \frac{1}{x^{2}} (1 - x^{2} y)$

Step 8

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

Tap for more steps...

Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

Differentiate.

Tap for more steps...

Step 8.2.1

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

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

Step 8.2.2

Since $\frac{1}{2} y^{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} y^{2}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x y] + \frac{d}{d x} [g (x)] = \frac{1}{x^{2}} (1 - x^{2} y)$

Step 8.3

Evaluate $\frac{d}{d x} [- x y]$ .

Tap for more steps...

Step 8.3.1

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

$0 - y \frac{d}{d x} [x] + \frac{d}{d x} [g (x)] = \frac{1}{x^{2}} (1 - x^{2} y)$

Step 8.3.2

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

$0 - y \cdot 1 + \frac{d}{d x} [g (x)] = \frac{1}{x^{2}} (1 - x^{2} y)$

Step 8.3.3

Multiply $- 1$ by $1$ .

$0 - y + \frac{d}{d x} [g (x)] = \frac{1}{x^{2}} (1 - x^{2} y)$

Step 8.4

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

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

Step 8.5

Simplify.

Tap for more steps...

Step 8.5.1

Subtract $y$ from $0$ .

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

Step 8.5.2

Reorder terms.

$\frac{d g}{d x} - y = \frac{1}{x^{2}} (1 - x^{2} y)$

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

Tap for more steps...

Step 9.1.1.1

Rewrite.

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

Step 9.1.1.2

Simplify by adding zeros.

$\frac{d g}{d x} - y = \frac{1}{x^{2}} (1 - x^{2} y)$

Step 9.1.1.3

Multiply $\frac{1}{x^{2}}$ by $1 - x^{2} y$ .

$\frac{d g}{d x} - y = \frac{1 - x^{2} y}{x^{2}}$

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

Add $y$ to both sides of the equation.

$\frac{d g}{d x} = \frac{1 - x^{2} y}{x^{2}} + y$

Step 9.1.2.2

Simplify each term.

Tap for more steps...

Step 9.1.2.2.1

Split the fraction $\frac{1 - x^{2} y}{x^{2}}$ into two fractions.

$\frac{d g}{d x} = \frac{1}{x^{2}} + \frac{- x^{2} y}{x^{2}} + y$

Step 9.1.2.2.2

Simplify each term.

Tap for more steps...

Step 9.1.2.2.2.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 9.1.2.2.2.1.1

Cancel the common factor.

$\frac{d g}{d x} = \frac{1}{x^{2}} + \frac{- x^{2} y}{x^{2}} + y$

Step 9.1.2.2.2.1.2

Divide $- 1 y$ by $1$ .

$\frac{d g}{d x} = \frac{1}{x^{2}} - 1 y + y$

Step 9.1.2.2.2.2

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

$\frac{d g}{d x} = \frac{1}{x^{2}} - y + y$

Step 9.1.2.3

Combine the opposite terms in $\frac{1}{x^{2}} - y + y$ .

Tap for more steps...

Step 9.1.2.3.1

Add $- y$ and $y$ .

$\frac{d g}{d x} = \frac{1}{x^{2}} + 0$

Step 9.1.2.3.2

Add $\frac{1}{x^{2}}$ and $0$ .

$\frac{d g}{d x} = \frac{1}{x^{2}}$

Step 10

Find the antiderivative of $\frac{1}{x^{2}}$ to find $g (x)$ .

Tap for more steps...

Step 10.1

Integrate both sides of $\frac{d g}{d x} = \frac{1}{x^{2}}$ .

$\int \frac{d g}{d x} d x = \int \frac{1}{x^{2}} d x$

Step 10.2

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

$g (x) = \int \frac{1}{x^{2}} d x$

Step 10.3

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$g (x) = \int {(x^{2})}^{- 1} d x$

Step 10.4

Multiply the exponents in ${(x^{2})}^{- 1}$ .

Tap for more steps...

Step 10.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$g (x) = \int x^{2 \cdot - 1} d x$

Step 10.4.2

Multiply $2$ by $- 1$ .

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

Step 10.5

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

$g (x) = - x^{- 1} + C$

Step 10.6

Rewrite $- x^{- 1} + C$ as $- \frac{1}{x} + C$ .

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

Step 11

Substitute for $g (x)$ in $f (x, y) = \frac{1}{2} y^{2} - x y + g (x)$ .

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

Step 12

Combine $\frac{1}{2}$ and $y^{2}$ .

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