Calculus Examples

$\frac{d y}{d x} = \frac{1}{3 x^{2} y^{2}}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{1}{3 x^{2}} \cdot \frac{1}{y^{2}}$

Step 1.2

Multiply both sides by $y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} (\frac{1}{3 x^{2}} \cdot \frac{1}{y^{2}})$

Step 1.3

Simplify.

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Step 1.3.1

Combine.

$y^{2} \frac{d y}{d x} = y^{2} \frac{1 \cdot 1}{3 x^{2} y^{2}}$

Step 1.3.2

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

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Step 1.3.2.1

Factor $y^{2}$ out of $3 x^{2} y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} \frac{1 \cdot 1}{y^{2} (3 x^{2})}$

Step 1.3.2.2

Cancel the common factor.

$y^{2} \frac{d y}{d x} = y^{2} \frac{1 \cdot 1}{y^{2} (3 x^{2})}$

Step 1.3.2.3

Rewrite the expression.

$y^{2} \frac{d y}{d x} = \frac{1 \cdot 1}{3 x^{2}}$

Step 1.3.3

Multiply $1$ by $1$ .

$y^{2} \frac{d y}{d x} = \frac{1}{3 x^{2}}$

Step 1.4

Rewrite the equation.

$y^{2} d y = \frac{1}{3 x^{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} d y = \int \frac{1}{3 x^{2}} d x$

Step 2.2

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

$\frac{1}{3} y^{3} + C_{1} = \int \frac{1}{3 x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{3} \int \frac{1}{x^{2}} d x$

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

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

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{3} \int {(x^{2})}^{- 1} d x$

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{3} \int x^{2 \cdot - 1} d x$

Step 2.3.2.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{3} \int x^{- 2} d x$

Step 2.3.3

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

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{3} (- x^{- 1} + C_{2})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $\frac{1}{3} (- x^{- 1} + C_{2})$ as $\frac{1}{3} (- \frac{1}{x}) + C_{2}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{1}{3} (- \frac{1}{x}) + C_{2}$

Step 2.3.4.2

Multiply $\frac{1}{3}$ by $\frac{1}{x}$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{3 x} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{3} y^{3} = - \frac{1}{3 x} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (- \frac{1}{3 x} + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $3 (\frac{1}{3} y^{3})$ .

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Step 3.2.1.1.1

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

$3 \frac{y^{3}}{3} = 3 (- \frac{1}{3 x} + K)$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$3 \frac{y^{3}}{3} = 3 (- \frac{1}{3 x} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (- \frac{1}{3 x} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (- \frac{1}{3 x} + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$y^{3} = 3 (- \frac{1}{3 x}) + 3 K$

Step 3.2.2.1.2

Cancel the common factor of $3$ .

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Step 3.2.2.1.2.1

Move the leading negative in $- \frac{1}{3 x}$ into the numerator.

$y^{3} = 3 \frac{- 1}{3 x} + 3 K$

Step 3.2.2.1.2.2

Factor $3$ out of $3 x$ .

$y^{3} = 3 \frac{- 1}{3 (x)} + 3 K$

Step 3.2.2.1.2.3

Cancel the common factor.

$y^{3} = 3 \frac{- 1}{3 x} + 3 K$

Step 3.2.2.1.2.4

Rewrite the expression.

$y^{3} = \frac{- 1}{x} + 3 K$

Step 3.2.2.1.3

Move the negative in front of the fraction.

$y^{3} = - \frac{1}{x} + 3 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{- \frac{1}{x} + 3 K}$

Step 3.4

Simplify $\sqrt[3]{- \frac{1}{x} + 3 K}$ .

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Step 3.4.1

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

$y = \sqrt[3]{- \frac{1}{x} + \frac{3 K x}{x}}$

Step 3.4.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{- 1 + 3 K x}{x}}$

Step 3.4.3

Rewrite $\sqrt[3]{\frac{- 1 + 3 K x}{x}}$ as $\frac{\sqrt[3]{- 1 + 3 K x}}{\sqrt[3]{x}}$ .

$y = \frac{\sqrt[3]{- 1 + 3 K x}}{\sqrt[3]{x}}$

Step 3.4.4

Multiply $\frac{\sqrt[3]{- 1 + 3 K x}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$y = \frac{\sqrt[3]{- 1 + 3 K x}}{\sqrt[3]{x}} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$

Step 3.4.5

Combine and simplify the denominator.

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Step 3.4.5.1

Multiply $\frac{\sqrt[3]{- 1 + 3 K x}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{\sqrt[3]{x} {\sqrt[3]{x}}^{2}}$

Step 3.4.5.2

Raise $\sqrt[3]{x}$ to the power of $1$ .

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{2}}$

Step 3.4.5.3

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

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1 + 2}}$

Step 3.4.5.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{3}}$

Step 3.4.5.5

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 3.4.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{{(x^{\frac{1}{3}})}^{3}}$

Step 3.4.5.5.2

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

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{x^{\frac{1}{3} \cdot 3}}$

Step 3.4.5.5.3

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

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 3.4.5.5.4

Cancel the common factor of $3$ .

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Step 3.4.5.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 3.4.5.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{x^{1}}$

Step 3.4.5.5.5

Simplify.

$y = \frac{\sqrt[3]{- 1 + 3 K x} {\sqrt[3]{x}}^{2}}{x}$

Step 3.4.6

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$y = \frac{\sqrt[3]{- 1 + 3 K x} \sqrt[3]{x^{2}}}{x}$

Step 3.4.7

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(- 1 + 3 K x) x^{2}}}{x}$

Step 3.4.8

Reorder factors in $\frac{\sqrt[3]{(- 1 + 3 K x) x^{2}}}{x}$ .

$y = \frac{\sqrt[3]{x^{2} (- 1 + 3 K x)}}{x}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{x^{2} (- 1 + K x)}}{x}$