Calculus Examples

$\frac{d y}{d x} = \frac{2 \sqrt{y}}{x}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\sqrt{y}}$ .

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

Step 1.2

Cancel the common factor of $\sqrt{y}$ .

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Step 1.2.1

Factor $\sqrt{y}$ out of $2 \sqrt{y}$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{2}{x}$

Step 1.3

Rewrite the equation.

$\frac{1}{\sqrt{y}} d y = \frac{2}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Multiply the exponents in ${(y^{\frac{1}{2}})}^{- 1}$ .

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Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

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

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

Step 2.2.1.3.3

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

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.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{\frac{1}{2}} = 2 \ln (| x |) + C$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{\frac{1}{2}} = 2 \ln (| x |) + C$ by $2$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

Divide $y^{\frac{1}{2}}$ by $1$ .

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Cancel the common factor of $2$ .

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Step 3.1.3.1.1

Cancel the common factor.

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

Step 3.1.3.1.2

Divide $\ln (| x |)$ by $1$ .

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

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1

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

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

Step 3.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

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

Step 3.3.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.2

Simplify.

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

Step 4

Simplify the constant of integration.

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