Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

Cancel the common factor.

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

Step 1.2.1.3

Rewrite the expression.

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

Step 1.2.2

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

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Move $y^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply basic rules of exponents.

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

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 $y^{3}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Apply the distributive property.

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

Step 3.2.2.1.2

Multiply $3 (- \frac{1}{x})$ .

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Step 3.2.2.1.2.1

Multiply $- 1$ by $3$ .

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

Step 3.2.2.1.2.2

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

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

Step 3.2.2.1.3

Move the negative in front of the fraction.

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

Factor $3$ out of $- \frac{3}{x} + 3 K$ .

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Step 3.4.1.1

Factor $3$ out of $- \frac{3}{x}$ .

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

Step 3.4.1.2

Factor $3$ out of $3 K$ .

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

Step 3.4.1.3

Factor $3$ out of $3 (- \frac{1}{x}) + 3 K$ .

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

Step 3.4.2

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

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

Step 3.4.3

Combine the numerators over the common denominator.

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

Step 3.4.4

Combine $3$ and $\frac{- 1 + K x}{x}$ .

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

Step 3.4.5

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

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

Step 3.4.6

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

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

Step 3.4.7

Combine and simplify the denominator.

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Step 3.4.7.1

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

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

Step 3.4.7.2

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

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

Step 3.4.7.3

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

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

Step 3.4.7.4

Add $1$ and $2$ .

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

Step 3.4.7.5

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

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Step 3.4.7.5.1

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

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

Step 3.4.7.5.2

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

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

Step 3.4.7.5.3

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

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

Step 3.4.7.5.4

Cancel the common factor of $3$ .

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Step 3.4.7.5.4.1

Cancel the common factor.

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

Step 3.4.7.5.4.2

Rewrite the expression.

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

Step 3.4.7.5.5

Simplify.

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

Step 3.4.8

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

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

Step 3.4.9

Combine using the product rule for radicals.

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

Step 3.4.10

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

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