Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \sqrt{y} d y = \int \sqrt{x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $\frac{3}{2}$ .

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

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 $\frac{3}{2} (\frac{2}{3} y^{\frac{3}{2}})$ .

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Step 3.2.1.1.1

Combine $\frac{2}{3}$ and $y^{\frac{3}{2}}$ .

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

Step 3.2.1.1.2

Combine.

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

Step 3.2.1.1.3

Cancel the common factor.

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

Step 3.2.1.1.4

Rewrite the expression.

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

Step 3.2.1.1.5

Cancel the common factor.

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

Step 3.2.1.1.6

Divide $y^{\frac{3}{2}}$ by $1$ .

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{3}{2} (\frac{2}{3} x^{\frac{3}{2}} + K)$ .

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Step 3.2.2.1.1

Simplify terms.

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Step 3.2.2.1.1.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

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

Step 3.2.2.1.1.2

Apply the distributive property.

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

Step 3.2.2.1.1.3

Combine.

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

Step 3.2.2.1.1.4

Combine $\frac{3}{2}$ and $K$ .

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

Step 3.2.2.1.2

Simplify each term.

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Step 3.2.2.1.2.1

Cancel the common factor.

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

Step 3.2.2.1.2.2

Rewrite the expression.

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

Step 3.2.2.1.2.3

Cancel the common factor.

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

Step 3.2.2.1.2.4

Divide $x^{\frac{3}{2}}$ by $1$ .

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

Step 3.3

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

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

Step 3.4

Simplify the left side.

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Step 3.4.1

Simplify ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 3.4.1.1

Multiply the exponents in ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 3.4.1.1.1

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

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

Step 3.4.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.1.1.2.1

Cancel the common factor.

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

Step 3.4.1.1.2.2

Rewrite the expression.

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

Step 3.4.1.1.3

Cancel the common factor of $2$ .

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Step 3.4.1.1.3.1

Cancel the common factor.

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

Step 3.4.1.1.3.2

Rewrite the expression.

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

Step 3.4.1.2

Simplify.

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

Step 4

Simplify the constant of integration.

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