Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Combine $\frac{x^{2}}{y^{2} - 2}$ and $\frac{d y}{d x}$ .

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

Step 1.1.2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

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

Step 1.1.3

Solve the equation for $\frac{d y}{d x}$ .

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Step 1.1.3.1

Simplify.

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Step 1.1.3.1.1

Remove parentheses.

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

Step 1.1.3.1.2

Multiply $y^{2} - 2$ by $1$ .

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

Step 1.1.3.2

Divide each term in $x^{2} \frac{d y}{d x} \cdot (2 y) = y^{2} - 2$ by $x^{2} \cdot 2 y$ and simplify.

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Step 1.1.3.2.1

Divide each term in $x^{2} \frac{d y}{d x} \cdot (2 y) = y^{2} - 2$ by $x^{2} \cdot 2 y$ .

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

Step 1.1.3.2.2

Simplify the left side.

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Step 1.1.3.2.2.1

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

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Step 1.1.3.2.2.1.1

Cancel the common factor.

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

Step 1.1.3.2.2.1.2

Rewrite the expression.

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

Step 1.1.3.2.2.2

Cancel the common factor of $2$ .

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Step 1.1.3.2.2.2.1

Cancel the common factor.

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

Step 1.1.3.2.2.2.2

Rewrite the expression.

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

Step 1.1.3.2.2.3

Cancel the common factor of $y$ .

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Step 1.1.3.2.2.3.1

Cancel the common factor.

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

Step 1.1.3.2.2.3.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.3.2.3

Simplify the right side.

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Step 1.1.3.2.3.1

Simplify each term.

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Step 1.1.3.2.3.1.1

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

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Step 1.1.3.2.3.1.1.1

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

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

Step 1.1.3.2.3.1.1.2

Cancel the common factors.

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Step 1.1.3.2.3.1.1.2.1

Factor $y$ out of $x^{2} \cdot 2 y$ .

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

Step 1.1.3.2.3.1.1.2.2

Cancel the common factor.

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

Step 1.1.3.2.3.1.1.2.3

Rewrite the expression.

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

Step 1.1.3.2.3.1.2

Move $2$ to the left of $x^{2}$ .

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

Step 1.1.3.2.3.1.3

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

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Step 1.1.3.2.3.1.3.1

Factor $2$ out of $- 2$ .

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

Step 1.1.3.2.3.1.3.2

Cancel the common factors.

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Step 1.1.3.2.3.1.3.2.1

Factor $2$ out of $x^{2} \cdot 2 y$ .

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

Step 1.1.3.2.3.1.3.2.2

Cancel the common factor.

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

Step 1.1.3.2.3.1.3.2.3

Rewrite the expression.

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

Step 1.1.3.2.3.1.4

Move the negative in front of the fraction.

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

Step 1.2

Factor.

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Step 1.2.1

To write $\frac{y}{2 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 1.2.2

To write $- \frac{1}{x^{2} y}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.2.3

Write each expression with a common denominator of $2 y x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.3.1

Multiply $\frac{y}{2 x^{2}}$ by $\frac{y}{y}$ .

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

Step 1.2.3.2

Multiply $\frac{1}{x^{2} y}$ by $\frac{2}{2}$ .

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

Step 1.2.3.3

Reorder the factors of $x^{2} y \cdot 2$ .

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Multiply $y$ by $y$ .

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

Step 1.3

Regroup factors.

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

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factor of $2 y$ .

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Step 1.5.2.1

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

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.5.3

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

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Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

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

Step 1.6

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

Let $u = y^{2} - 2$ . Then $d u = 2 y d y$ , so $\frac{1}{2} d u = y d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = y^{2} - 2$ . Find $\frac{d u}{d y}$ .

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Step 2.2.2.1.1

Differentiate $y^{2} - 2$ .

$\frac{d}{d y} [y^{2} - 2]$

Step 2.2.2.1.2

By the Sum Rule, the derivative of $y^{2} - 2$ with respect to $y$ is $\frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 2]$ .

$\frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 2]$

Step 2.2.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$2 y + \frac{d}{d y} [- 2]$

Step 2.2.2.1.4

Since $- 2$ is constant with respect to $y$ , the derivative of $- 2$ with respect to $y$ is $0$ .

$2 y + 0$

Step 2.2.2.1.5

Add $2 y$ and $0$ .

$2 y$

Step 2.2.2.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

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

Step 2.2.3.2

Move $2$ to the left of $u$ .

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

Step 2.2.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$2 (\frac{1}{2} \int \frac{1}{u} d u) = \int \frac{1}{x^{2}} d x$

Step 2.2.5

Simplify.

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Step 2.2.5.1

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

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

Step 2.2.5.2

Cancel the common factor of $2$ .

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Step 2.2.5.2.1

Cancel the common factor.

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

Step 2.2.5.2.2

Rewrite the expression.

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

Step 2.2.5.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

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

Step 2.2.6

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

$\ln (| u |) + C_{1} = \int \frac{1}{x^{2}} d x$

Step 2.2.7

Replace all occurrences of $u$ with $y^{2} - 2$ .

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

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.

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

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, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (| y^{2} - 2 |) + C_{1} = \int x^{2 \cdot - 1} d x$

Step 2.3.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.2

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

$\ln (| y^{2} - 2 |) + C_{1} = - x^{- 1} + C_{2}$

Step 2.3.3

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

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

Step 2.4

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

$\ln (| y^{2} - 2 |) = - \frac{1}{x} + C$

Step 3

Solve for $y$ .

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Step 3.1

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y^{2} - 2 |)} = e^{- \frac{1}{x} + C}$

Step 3.2

Rewrite $\ln (| y^{2} - 2 |) = - \frac{1}{x} + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- \frac{1}{x} + C} = | y^{2} - 2 |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y^{2} - 2 | = e^{- \frac{1}{x} + C}$ .

$| y^{2} - 2 | = e^{- \frac{1}{x} + C}$

Step 3.3.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y^{2} - 2 = \pm e^{- \frac{1}{x} + C}$

Step 3.3.3

Add $2$ to both sides of the equation.

$y^{2} = \pm e^{- \frac{1}{x} + C} + 2$

Step 3.3.4

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

$y = \pm \sqrt{\pm e^{- \frac{1}{x} + C} + 2}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{- \frac{1}{x} + C}$ as $e^{- \frac{1}{x}} e^{C}$ .

$y = \pm \sqrt{\pm e^{- \frac{1}{x}} e^{C} + 2}$

Step 4.2

Reorder $e^{- \frac{1}{x}}$ and $e^{C}$ .

$y = \pm \sqrt{\pm e^{C} e^{- \frac{1}{x}} + 2}$

Step 4.3

Combine constants with the plus or minus.

$y = \pm \sqrt{C e^{- \frac{1}{x}} + 2}$