Calculus Examples

$(4 y - 3 x) d x + 5 x d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 4 y - 3 x$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [4 y - 3 x]$

Step 1.2

By the Sum Rule, the derivative of $4 y - 3 x$ with respect to $y$ is $\frac{d}{d y} [4 y] + \frac{d}{d y} [- 3 x]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [4 y] + \frac{d}{d y} [- 3 x]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 4 \frac{d}{d y} [y] + \frac{d}{d y} [- 3 x]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 4 \cdot 1 + \frac{d}{d y} [- 3 x]$

Step 1.3.3

Multiply $4$ by $1$ .

$\frac{\partial M}{\partial y} = 4 + \frac{d}{d y} [- 3 x]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $- 3 x$ is constant with respect to $y$ , the derivative of $- 3 x$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 4 + 0$

Step 1.4.2

Add $4$ and $0$ .

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

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = 5 x$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [5 x]$

Step 2.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{\partial N}{\partial x} = 5 \frac{d}{d x} [x]$

Step 2.3

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

$\frac{\partial N}{\partial x} = 5 \cdot 1$

Step 2.4

Multiply $5$ by $1$ .

$\frac{\partial N}{\partial x} = 5$

Step 3

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

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Step 3.1

Substitute $4$ for $\frac{\partial M}{\partial y}$ and $5$ for $\frac{\partial N}{\partial x}$ .

$4 = 5$

Step 3.2

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

$4 = 5$ is not an identity.

Step 4

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

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Step 4.1

Substitute $4$ for $\frac{\partial M}{\partial y}$ .

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

Step 4.2

Substitute $5$ for $\frac{\partial N}{\partial x}$ .

$\frac{4 - 5}{N}$

Step 4.3

Substitute $5 x$ for $N$ .

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Step 4.3.1

Substitute $5 x$ for $N$ .

$\frac{4 - 5}{5 x}$

Step 4.3.2

Subtract $5$ from $4$ .

$\frac{- 1}{5 x}$

Step 4.3.3

Move the negative in front of the fraction.

$- \frac{1}{5 x}$

Step 4.4

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

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

Step 5

Evaluate the integral $e^{\int - \frac{1}{5 x} d x}$ .

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Step 5.1

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

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

Step 5.2

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

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

Step 5.3

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

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

Step 5.4

Simplify.

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

Step 5.5

Simplify each term.

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Step 5.5.1

Multiply $- \frac{1}{5} \ln (| x |)$ .

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Step 5.5.1.1

Reorder $\ln (| x |)$ and $\frac{1}{5}$ .

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

Step 5.5.1.2

Simplify $\frac{1}{5} \ln (| x |)$ by moving $\frac{1}{5}$ inside the logarithm.

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

Step 5.5.2

Simplify $- \ln ({| x |}^{\frac{1}{5}})$ by moving $- 1$ inside the logarithm.

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

Step 5.5.3

Exponentiation and log are inverse functions.

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

Step 5.5.4

Multiply the exponents in ${({| x |}^{\frac{1}{5}})}^{- 1}$ .

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Step 5.5.4.1

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

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

Step 5.5.4.2

Combine $\frac{1}{5}$ and $- 1$ .

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

Step 5.5.4.3

Move the negative in front of the fraction.

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

Step 5.5.5

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

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

Step 6

Multiply both sides of $(4 y - 3 x) d x + 5 x d y = 0$ by the integration factor $\frac{1}{x^{\frac{1}{5}}}$ .

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Step 6.1

Multiply $4 y - 3 x$ by $\frac{1}{x^{\frac{1}{5}}}$ .

$(4 y - 3 x) \frac{1}{x^{\frac{1}{5}}} d x + 5 x d y = 0$

Step 6.2

Multiply $4 y - 3 x$ by $\frac{1}{x^{\frac{1}{5}}}$ .

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + 5 x d y = 0$

Step 6.3

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

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + 5 x \frac{1}{x^{\frac{1}{5}}} d y = 0$

Step 6.4

Multiply $5 x \frac{1}{x^{\frac{1}{5}}}$ .

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Step 6.4.1

Combine $5$ and $\frac{1}{x^{\frac{1}{5}}}$ .

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + x \frac{5}{x^{\frac{1}{5}}} d y = 0$

Step 6.4.2

Combine $x$ and $\frac{5}{x^{\frac{1}{5}}}$ .

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + \frac{x \cdot 5}{x^{\frac{1}{5}}} d y = 0$

Step 6.5

Move $x^{\frac{1}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + x \cdot 5 x^{- \frac{1}{5}} d y = 0$

Step 6.6

Multiply $x$ by $x^{- \frac{1}{5}}$ by adding the exponents.

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Step 6.6.1

Move $x^{- \frac{1}{5}}$ .

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + x^{- \frac{1}{5}} x \cdot 5 d y = 0$

Step 6.6.2

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

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Step 6.6.2.1

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

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + x^{- \frac{1}{5}} x^{1} \cdot 5 d y = 0$

Step 6.6.2.2

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

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + x^{- \frac{1}{5} + 1} \cdot 5 d y = 0$

Step 6.6.3

Write $1$ as a fraction with a common denominator.

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + x^{- \frac{1}{5} + \frac{5}{5}} \cdot 5 d y = 0$

Step 6.6.4

Combine the numerators over the common denominator.

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + x^{\frac{- 1 + 5}{5}} \cdot 5 d y = 0$

Step 6.6.5

Add $- 1$ and $5$ .

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + x^{\frac{4}{5}} \cdot 5 d y = 0$

Step 6.7

Move $5$ to the left of $x^{\frac{4}{5}}$ .

$\frac{4 y - 3 x}{x^{\frac{1}{5}}} d x + 5 x^{\frac{4}{5}} d y = 0$

Step 7

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

$f (x, y) = \int 5 x^{\frac{4}{5}} d y$

Step 8

Integrate $N (x, y) = 5 x^{\frac{4}{5}}$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

$f (x, y) = 5 x^{\frac{4}{5}} y + C$

Step 9

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

$f (x, y) = 5 x^{\frac{4}{5}} y + g (x)$

Step 10

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

$\frac{\partial f}{\partial x} = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [5 x^{\frac{4}{5}} y + g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.2

By the Sum Rule, the derivative of $5 x^{\frac{4}{5}} y + g (x)$ with respect to $x$ is $\frac{d}{d x} [5 x^{\frac{4}{5}} y] + \frac{d}{d x} [g (x)]$ .

$\frac{d}{d x} [5 x^{\frac{4}{5}} y] + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3

Evaluate $\frac{d}{d x} [5 x^{\frac{4}{5}} y]$ .

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Step 11.3.1

Since $5 y$ is constant with respect to $x$ , the derivative of $5 x^{\frac{4}{5}} y$ with respect to $x$ is $5 y \frac{d}{d x} [x^{\frac{4}{5}}]$ .

$5 y \frac{d}{d x} [x^{\frac{4}{5}}] + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{5}$ .

$5 y (\frac{4}{5} x^{\frac{4}{5} - 1}) + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$5 y (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}}) + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.4

Combine $- 1$ and $\frac{5}{5}$ .

$5 y (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}}) + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.5

Combine the numerators over the common denominator.

$5 y (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}}) + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.6

Simplify the numerator.

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Step 11.3.6.1

Multiply $- 1$ by $5$ .

$5 y (\frac{4}{5} x^{\frac{4 - 5}{5}}) + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.6.2

Subtract $5$ from $4$ .

$5 y (\frac{4}{5} x^{\frac{- 1}{5}}) + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.7

Move the negative in front of the fraction.

$5 y (\frac{4}{5} x^{- \frac{1}{5}}) + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.8

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

$5 y \frac{4 x^{- \frac{1}{5}}}{5} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.9

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

$y \frac{5 (4 x^{- \frac{1}{5}})}{5} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.10

Multiply $4$ by $5$ .

$y \frac{20 x^{- \frac{1}{5}}}{5} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.11

Combine $y$ and $\frac{20 x^{- \frac{1}{5}}}{5}$ .

$\frac{y (20 x^{- \frac{1}{5}})}{5} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.12

Move $20$ to the left of $y$ .

$\frac{20 \cdot y x^{- \frac{1}{5}}}{5} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.13

Move $x^{- \frac{1}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{20 y}{5 x^{\frac{1}{5}}} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.14

Factor $5$ out of $20 y$ .

$\frac{5 (4 y)}{5 x^{\frac{1}{5}}} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.15

Cancel the common factors.

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Step 11.3.15.1

Factor $5$ out of $5 x^{\frac{1}{5}}$ .

$\frac{5 (4 y)}{5 (x^{\frac{1}{5}})} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.15.2

Cancel the common factor.

$\frac{5 (4 y)}{5 x^{\frac{1}{5}}} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.3.15.3

Rewrite the expression.

$\frac{4 y}{x^{\frac{1}{5}}} + \frac{d}{d x} [g (x)] = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.4

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

$\frac{4 y}{x^{\frac{1}{5}}} + \frac{d g}{d x} = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 11.5

Reorder terms.

$\frac{d g}{d x} + \frac{4 y}{x^{\frac{1}{5}}} = \frac{4 y - 3 x}{x^{\frac{1}{5}}}$

Step 12

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

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Step 12.1

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

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Step 12.1.1

Simplify $\frac{d g}{d x} + \frac{4 y}{x^{\frac{1}{5}}} - \frac{4 y - 3 x}{x^{\frac{1}{5}}}$ .

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Step 12.1.1.1

Combine the numerators over the common denominator.

$\frac{d g}{d x} + \frac{4 y - (4 y - 3 x)}{x^{\frac{1}{5}}} = 0$

Step 12.1.1.2

Simplify each term.

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Step 12.1.1.2.1

Apply the distributive property.

$\frac{d g}{d x} + \frac{4 y - (4 y) - (- 3 x)}{x^{\frac{1}{5}}} = 0$

Step 12.1.1.2.2

Multiply $4$ by $- 1$ .

$\frac{d g}{d x} + \frac{4 y - 4 y - (- 3 x)}{x^{\frac{1}{5}}} = 0$

Step 12.1.1.2.3

Multiply $- 3$ by $- 1$ .

$\frac{d g}{d x} + \frac{4 y - 4 y + 3 x}{x^{\frac{1}{5}}} = 0$

Step 12.1.1.3

Simplify by adding terms.

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Step 12.1.1.3.1

Subtract $4 y$ from $4 y$ .

$\frac{d g}{d x} + \frac{0 + 3 x}{x^{\frac{1}{5}}} = 0$

Step 12.1.1.3.2

Add $0$ and $3 x$ .

$\frac{d g}{d x} + \frac{3 x}{x^{\frac{1}{5}}} = 0$

Step 12.1.1.4

Simplify each term.

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Step 12.1.1.4.1

Move $x^{\frac{1}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{d g}{d x} + 3 x \cdot x^{- \frac{1}{5}} = 0$

Step 12.1.1.4.2

Multiply $x$ by $x^{- \frac{1}{5}}$ by adding the exponents.

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Step 12.1.1.4.2.1

Move $x^{- \frac{1}{5}}$ .

$\frac{d g}{d x} + 3 (x^{- \frac{1}{5}} x) = 0$

Step 12.1.1.4.2.2

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

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Step 12.1.1.4.2.2.1

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

$\frac{d g}{d x} + 3 (x^{- \frac{1}{5}} x^{1}) = 0$

Step 12.1.1.4.2.2.2

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

$\frac{d g}{d x} + 3 x^{- \frac{1}{5} + 1} = 0$

Step 12.1.1.4.2.3

Write $1$ as a fraction with a common denominator.

$\frac{d g}{d x} + 3 x^{- \frac{1}{5} + \frac{5}{5}} = 0$

Step 12.1.1.4.2.4

Combine the numerators over the common denominator.

$\frac{d g}{d x} + 3 x^{\frac{- 1 + 5}{5}} = 0$

Step 12.1.1.4.2.5

Add $- 1$ and $5$ .

$\frac{d g}{d x} + 3 x^{\frac{4}{5}} = 0$

Step 12.1.2

Subtract $3 x^{\frac{4}{5}}$ from both sides of the equation.

$\frac{d g}{d x} = - 3 x^{\frac{4}{5}}$

Step 13

Find the antiderivative of $- 3 x^{\frac{4}{5}}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = - 3 x^{\frac{4}{5}}$ .

$\int \frac{d g}{d x} d x = \int - 3 x^{\frac{4}{5}} d x$

Step 13.2

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

$g (x) = \int - 3 x^{\frac{4}{5}} d x$

Step 13.3

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$g (x) = - 3 \int x^{\frac{4}{5}} d x$

Step 13.4

By the Power Rule, the integral of $x^{\frac{4}{5}}$ with respect to $x$ is $\frac{5}{9} x^{\frac{9}{5}}$ .

$g (x) = - 3 (\frac{5}{9} x^{\frac{9}{5}} + C)$

Step 13.5

Simplify the answer.

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Step 13.5.1

Rewrite $- 3 (\frac{5}{9} x^{\frac{9}{5}} + C)$ as $- 3 (\frac{5}{9}) x^{\frac{9}{5}} + C$ .

$g (x) = - 3 (\frac{5}{9}) x^{\frac{9}{5}} + C$

Step 13.5.2

Simplify.

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Step 13.5.2.1

Combine $- 3$ and $\frac{5}{9}$ .

$g (x) = \frac{- 3 \cdot 5}{9} x^{\frac{9}{5}} + C$

Step 13.5.2.2

Multiply $- 3$ by $5$ .

$g (x) = \frac{- 15}{9} x^{\frac{9}{5}} + C$

Step 13.5.2.3

Cancel the common factor of $- 15$ and $9$ .

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Step 13.5.2.3.1

Factor $3$ out of $- 15$ .

$g (x) = \frac{3 (- 5)}{9} x^{\frac{9}{5}} + C$

Step 13.5.2.3.2

Cancel the common factors.

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Step 13.5.2.3.2.1

Factor $3$ out of $9$ .

$g (x) = \frac{3 \cdot - 5}{3 \cdot 3} x^{\frac{9}{5}} + C$

Step 13.5.2.3.2.2

Cancel the common factor.

$g (x) = \frac{3 \cdot - 5}{3 \cdot 3} x^{\frac{9}{5}} + C$

Step 13.5.2.3.2.3

Rewrite the expression.

$g (x) = \frac{- 5}{3} x^{\frac{9}{5}} + C$

Step 13.5.2.4

Move the negative in front of the fraction.

$g (x) = - \frac{5}{3} x^{\frac{9}{5}} + C$

Step 14

Substitute for $g (x)$ in $f (x, y) = 5 x^{\frac{4}{5}} y + g (x)$ .

$f (x, y) = 5 x^{\frac{4}{5}} y - \frac{5}{3} x^{\frac{9}{5}} + C$

Step 15

Simplify $f (x, y) = 5 x^{\frac{4}{5}} y - \frac{5}{3} x^{\frac{9}{5}} + C$ .

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Step 15.1

Simplify each term.

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Step 15.1.1

Combine $x^{\frac{9}{5}}$ and $\frac{5}{3}$ .

$f (x, y) = 5 x^{\frac{4}{5}} y - \frac{x^{\frac{9}{5}} \cdot 5}{3} + C$

Step 15.1.2

Move $5$ to the left of $x^{\frac{9}{5}}$ .

$f (x, y) = 5 x^{\frac{4}{5}} y - \frac{5 x^{\frac{9}{5}}}{3} + C$

Step 15.2

Reorder factors in $f (x, y) = 5 x^{\frac{4}{5}} y - \frac{5 x^{\frac{9}{5}}}{3} + C$ .

$f (x, y) = 5 y x^{\frac{4}{5}} - \frac{5 x^{\frac{9}{5}}}{3} + C$