Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $y^{2}$ by $y^{- 2}$ by adding the exponents.

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Step 1.2.3.1

Move $y^{- 2}$ .

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

Step 1.2.3.2

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

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

Step 1.2.3.3

Add $- 2$ and $2$ .

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

Step 1.2.4

Simplify $2 y^{0} x$ .

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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 $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

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

Step 2.3.3.2.2.2

Rewrite the expression.

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

Step 2.3.3.2.3

Multiply $x^{2}$ by $1$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

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 $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (x^{2} + K)$

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.

$3 \frac{y^{3}}{3} = 3 (x^{2} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (x^{2} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$y^{3} = 3 x^{2} + 3 K$

Step 3.3

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

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

Step 3.4

Factor $3$ out of $3 x^{2} + 3 K$ .

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Step 3.4.1

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

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

Step 3.4.2

Factor $3$ out of $3 (x^{2}) + 3 K$ .

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