Calculus Examples

$y d x - 4 (x + y^{6}) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = - 4 (x + y^{6})$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [- 4 (x + y^{6})]$

Step 2.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 (x + y^{6})$ with respect to $x$ is $- 4 \frac{d}{d x} [x + y^{6}]$ .

$\frac{\partial N}{\partial x} = - 4 \frac{d}{d x} [x + y^{6}]$

Step 2.3

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

$\frac{\partial N}{\partial x} = - 4 (\frac{d}{d x} [x] + \frac{d}{d x} [y^{6}])$

Step 2.4

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

$\frac{\partial N}{\partial x} = - 4 (1 + \frac{d}{d x} [y^{6}])$

Step 2.5

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

$\frac{\partial N}{\partial x} = - 4 (1 + 0)$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $1$ and $0$ .

$\frac{\partial N}{\partial x} = - 4 \cdot 1$

Step 2.6.2

Multiply $- 4$ by $1$ .

$\frac{\partial N}{\partial x} = - 4$

Step 3

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

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Step 3.1

Substitute $1$ for $\frac{\partial M}{\partial y}$ and $- 4$ for $\frac{\partial N}{\partial x}$ .

$1 = - 4$

Step 3.2

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

$1 = - 4$ is not an identity.

Step 4

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

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Step 4.1

Substitute $- 4$ for $\frac{\partial N}{\partial x}$ .

$\frac{- 4 - \frac{\partial M}{\partial y}}{M}$

Step 4.2

Substitute $1$ for $\frac{\partial M}{\partial y}$ .

$\frac{- 4 - 1}{M}$

Step 4.3

Substitute $y$ for $M$ .

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Step 4.3.1

Substitute $y$ for $M$ .

$\frac{- 4 - 1}{y}$

Step 4.3.2

Subtract $1$ from $- 4$ .

$\frac{- 5}{y}$

Step 4.3.3

Substitute $y$ for $M$ .

$- \frac{5}{y}$

Step 4.4

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

$μ (x, y) = e^{\int - \frac{5}{y} d y}$

Step 5

Evaluate the integral $e^{\int - \frac{5}{y} d y}$ .

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Step 5.1

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

$μ (x, y) = e^{- \int \frac{5}{y} d y}$

Step 5.2

Since $5$ is constant with respect to $y$ , move $5$ out of the integral.

$μ (x, y) = e^{- (5 \int \frac{1}{y} d y)}$

Step 5.3

Multiply $5$ by $- 1$ .

$μ (x, y) = e^{- 5 \int \frac{1}{y} d y}$

Step 5.4

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$μ (x, y) = e^{- 5 (\ln (| y |) + C)}$

Step 5.5

Simplify.

$μ (x, y) = e^{- 5 \ln (| y |)} + C$

Step 5.6

Simplify each term.

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Step 5.6.1

Simplify $- 5 \ln (| y |)$ by moving $- 5$ inside the logarithm.

$μ (x, y) = e^{\ln ({| y |}^{- 5})} + C$

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| y |}^{- 5} + C$

Step 5.6.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$μ (x, y) = \frac{1}{{| y |}^{5}} + C$

Step 6

Multiply both sides of $y d x - 4 (x + y^{6}) d y = 0$ by the integration factor $\frac{1}{y^{5}}$ .

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Step 6.1

Multiply $y$ by $\frac{1}{y^{5}}$ .

$y \frac{1}{y^{5}} d x - 4 (x + y^{6}) d y = 0$

Step 6.2

Cancel the common factor of $y$ .

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Step 6.2.1

Factor $y$ out of $y^{5}$ .

$y \frac{1}{y \cdot y^{4}} d x - 4 (x + y^{6}) d y = 0$

Step 6.2.2

Cancel the common factor.

$y \frac{1}{y \cdot y^{4}} d x - 4 (x + y^{6}) d y = 0$

Step 6.2.3

Rewrite the expression.

$\frac{1}{y^{4}} d x - 4 (x + y^{6}) d y = 0$

Step 6.3

Multiply $- 4 (x + y^{6})$ by $\frac{1}{y^{5}}$ .

$\frac{1}{y^{4}} d x - 4 (x + y^{6}) \frac{1}{y^{5}} d y = 0$

Step 6.4

Apply the distributive property.

$\frac{1}{y^{4}} d x + (- 4 x - 4 y^{6}) \frac{1}{y^{5}} d y = 0$

Step 6.5

Multiply $- 4 x - 4 y^{6}$ by $\frac{1}{y^{5}}$ .

$\frac{1}{y^{4}} d x + \frac{- 4 x - 4 y^{6}}{y^{5}} d y = 0$

Step 6.6

Factor $4$ out of $- 4 x - 4 y^{6}$ .

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Step 6.6.1

Factor $4$ out of $- 4 x$ .

$\frac{1}{y^{4}} d x + \frac{4 (- x) - 4 y^{6}}{y^{5}} d y = 0$

Step 6.6.2

Factor $4$ out of $- 4 y^{6}$ .

$\frac{1}{y^{4}} d x + \frac{4 (- x) + 4 (- y^{6})}{y^{5}} d y = 0$

Step 6.6.3

Factor $4$ out of $4 (- x) + 4 (- y^{6})$ .

$\frac{1}{y^{4}} d x + \frac{4 (- x - y^{6})}{y^{5}} d y = 0$

Step 6.7

Factor $- 1$ out of $- x$ .

$\frac{1}{y^{4}} d x + \frac{4 (- (x) - y^{6})}{y^{5}} d y = 0$

Step 6.8

Factor $- 1$ out of $- y^{6}$ .

$\frac{1}{y^{4}} d x + \frac{4 (- (x) - (y^{6}))}{y^{5}} d y = 0$

Step 6.9

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

$\frac{1}{y^{4}} d x + \frac{4 (- (x + y^{6}))}{y^{5}} d y = 0$

Step 6.10

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

$\frac{1}{y^{4}} d x + \frac{4 (- 1 (x + y^{6}))}{y^{5}} d y = 0$

Step 6.11

Move the negative in front of the fraction.

$\frac{1}{y^{4}} d x - \frac{4 (x + y^{6})}{y^{5}} d y = 0$

Step 7

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

$f (x, y) = \int \frac{1}{y^{4}} d x$

Step 8

Integrate $M (x, y) = \frac{1}{y^{4}}$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

$f (x, y) = \frac{1}{y^{4}} x + C$

Step 8.2

Combine $\frac{1}{y^{4}}$ and $x$ .

$f (x, y) = \frac{x}{y^{4}} + C$

Step 9

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

$f (x, y) = \frac{x}{y^{4}} + g (y)$

Step 10

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

$\frac{\partial f}{\partial y} = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [\frac{x}{y^{4}} + g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.2

By the Sum Rule, the derivative of $\frac{x}{y^{4}} + g (y)$ with respect to $y$ is $\frac{d}{d y} [\frac{x}{y^{4}}] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [\frac{x}{y^{4}}] + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3

Evaluate $\frac{d}{d y} [\frac{x}{y^{4}}]$ .

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Step 11.3.1

Since $x$ is constant with respect to $y$ , the derivative of $\frac{x}{y^{4}}$ with respect to $y$ is $x \frac{d}{d y} [\frac{1}{y^{4}}]$ .

$x \frac{d}{d y} [\frac{1}{y^{4}}] + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.2

Rewrite $\frac{1}{y^{4}}$ as ${(y^{4})}^{- 1}$ .

$x \frac{d}{d y} [{(y^{4})}^{- 1}] + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{- 1}$ and $g (y) = y^{4}$ .

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Step 11.3.3.1

To apply the Chain Rule, set $u$ as $y^{4}$ .

$x (\frac{d}{d u} [u^{- 1}] \frac{d}{d y} [y^{4}]) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.3.2

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

$x (- u^{- 2} \frac{d}{d y} [y^{4}]) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.3.3

Replace all occurrences of $u$ with $y^{4}$ .

$x (- {(y^{4})}^{- 2} \frac{d}{d y} [y^{4}]) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.4

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 4$ .

$x (- {(y^{4})}^{- 2} (4 y^{3})) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.5

Multiply the exponents in ${(y^{4})}^{- 2}$ .

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Step 11.3.5.1

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

$x (- y^{4 \cdot - 2} (4 y^{3})) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.5.2

Multiply $4$ by $- 2$ .

$x (- y^{- 8} (4 y^{3})) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.6

Multiply $4$ by $- 1$ .

$x (- 4 y^{- 8} y^{3}) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.7

Multiply $y^{- 8}$ by $y^{3}$ by adding the exponents.

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Step 11.3.7.1

Move $y^{3}$ .

$x (- 4 (y^{3} y^{- 8})) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.7.2

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

$x (- 4 y^{3 - 8}) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.3.7.3

Subtract $8$ from $3$ .

$x (- 4 y^{- 5}) + \frac{d}{d y} [g (y)] = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.4

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

$x (- 4 y^{- 5}) + \frac{d g}{d y} = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.5

Simplify.

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Step 11.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (- 4 \frac{1}{y^{5}}) + \frac{d g}{d y} = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.5.2

Combine terms.

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Step 11.5.2.1

Combine $- 4$ and $\frac{1}{y^{5}}$ .

$x \frac{- 4}{y^{5}} + \frac{d g}{d y} = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.5.2.2

Move the negative in front of the fraction.

$x (- \frac{4}{y^{5}}) + \frac{d g}{d y} = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.5.2.3

Combine $x$ and $\frac{4}{y^{5}}$ .

$- \frac{x \cdot 4}{y^{5}} + \frac{d g}{d y} = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.5.2.4

Move $4$ to the left of $x$ .

$- \frac{4 x}{y^{5}} + \frac{d g}{d y} = - \frac{4 (x + y^{6})}{y^{5}}$

Step 11.5.3

Reorder terms.

$\frac{d g}{d y} - \frac{4 x}{y^{5}} = - \frac{4 (x + y^{6})}{y^{5}}$

Step 12

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

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Step 12.1

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

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Step 12.1.1

Move all terms containing variables to the left side of the equation.

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Step 12.1.1.1

Add $\frac{4 (x + y^{6})}{y^{5}}$ to both sides of the equation.

$\frac{d g}{d y} - \frac{4 x}{y^{5}} + \frac{4 (x + y^{6})}{y^{5}} = 0$

Step 12.1.1.2

Combine the numerators over the common denominator.

$\frac{d g}{d y} + \frac{- 4 x + 4 (x + y^{6})}{y^{5}} = 0$

Step 12.1.1.3

Apply the distributive property.

$\frac{d g}{d y} + \frac{- 4 x + 4 x + 4 y^{6}}{y^{5}} = 0$

Step 12.1.1.4

Add $- 4 x$ and $4 x$ .

$\frac{d g}{d y} + \frac{0 + 4 y^{6}}{y^{5}} = 0$

Step 12.1.1.5

Add $0$ and $4 y^{6}$ .

$\frac{d g}{d y} + \frac{4 y^{6}}{y^{5}} = 0$

Step 12.1.1.6

Cancel the common factor of $y^{6}$ and $y^{5}$ .

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Step 12.1.1.6.1

Factor $y^{5}$ out of $4 y^{6}$ .

$\frac{d g}{d y} + \frac{y^{5} (4 y)}{y^{5}} = 0$

Step 12.1.1.6.2

Cancel the common factors.

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Step 12.1.1.6.2.1

Multiply by $1$ .

$\frac{d g}{d y} + \frac{y^{5} (4 y)}{y^{5} \cdot 1} = 0$

Step 12.1.1.6.2.2

Cancel the common factor.

$\frac{d g}{d y} + \frac{y^{5} (4 y)}{y^{5} \cdot 1} = 0$

Step 12.1.1.6.2.3

Rewrite the expression.

$\frac{d g}{d y} + \frac{4 y}{1} = 0$

Step 12.1.1.6.2.4

Divide $4 y$ by $1$ .

$\frac{d g}{d y} + 4 y = 0$

Step 12.1.2

Subtract $4 y$ from both sides of the equation.

$\frac{d g}{d y} = - 4 y$

Step 13

Find the antiderivative of $- 4 y$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = - 4 y$ .

$\int \frac{d g}{d y} d y = \int - 4 y d y$

Step 13.2

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

$g (y) = \int - 4 y d y$

Step 13.3

Since $- 4$ is constant with respect to $y$ , move $- 4$ out of the integral.

$g (y) = - 4 \int y d y$

Step 13.4

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

$g (y) = - 4 (\frac{1}{2} y^{2} + C)$

Step 13.5

Simplify the answer.

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Step 13.5.1

Rewrite $- 4 (\frac{1}{2} y^{2} + C)$ as $- 4 (\frac{1}{2}) y^{2} + C$ .

$g (y) = - 4 (\frac{1}{2}) y^{2} + C$

Step 13.5.2

Simplify.

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Step 13.5.2.1

Combine $- 4$ and $\frac{1}{2}$ .

$g (y) = \frac{- 4}{2} y^{2} + C$

Step 13.5.2.2

Cancel the common factor of $- 4$ and $2$ .

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Step 13.5.2.2.1

Factor $2$ out of $- 4$ .

$g (y) = \frac{2 \cdot - 2}{2} y^{2} + C$

Step 13.5.2.2.2

Cancel the common factors.

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Step 13.5.2.2.2.1

Factor $2$ out of $2$ .

$g (y) = \frac{2 \cdot - 2}{2 (1)} y^{2} + C$

Step 13.5.2.2.2.2

Cancel the common factor.

$g (y) = \frac{2 \cdot - 2}{2 \cdot 1} y^{2} + C$

Step 13.5.2.2.2.3

Rewrite the expression.

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

Step 13.5.2.2.2.4

Divide $- 2$ by $1$ .

$g (y) = - 2 y^{2} + C$

Step 14

Substitute for $g (y)$ in $f (x, y) = \frac{x}{y^{4}} + g (y)$ .

$f (x, y) = \frac{x}{y^{4}} - 2 y^{2} + C$