Calculus Examples

$\frac{d y}{d x} = - \frac{\sqrt{x}}{4 y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y (- \frac{\sqrt{x}}{4 y})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$y \frac{d y}{d x} = - y \frac{\sqrt{x}}{4 y}$

Step 1.2.2

Cancel the common factor of $y$ .

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Step 1.2.2.1

Factor $y$ out of $- y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{\sqrt{x}}{4 y}$

Step 1.2.2.2

Factor $y$ out of $4 y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{\sqrt{x}}{y \cdot 4}$

Step 1.2.2.3

Cancel the common factor.

$y \frac{d y}{d x} = y \cdot - 1 \frac{\sqrt{x}}{y \cdot 4}$

Step 1.2.2.4

Rewrite the expression.

$y \frac{d y}{d x} = - \frac{\sqrt{x}}{4}$

Step 1.3

Rewrite the equation.

$y d y = - \frac{\sqrt{x}}{4} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int - \frac{\sqrt{x}}{4} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int - \frac{\sqrt{x}}{4} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = - \int \frac{\sqrt{x}}{4} d x$

Step 2.3.2

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = - (\frac{1}{4} \int \sqrt{x} d x)$

Step 2.3.3

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

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{4} \int x^{\frac{1}{2}} d x$

Step 2.3.4

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

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{4} (\frac{2}{3} x^{\frac{3}{2}} + C_{2})$

Step 2.3.5

Simplify the answer.

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Step 2.3.5.1

Rewrite $- \frac{1}{4} (\frac{2}{3} x^{\frac{3}{2}} + C_{2})$ as $- \frac{1}{4} \cdot \frac{2}{3} x^{\frac{3}{2}} + C_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{4} \cdot \frac{2}{3} x^{\frac{3}{2}} + C_{2}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Multiply $\frac{2}{3}$ by $\frac{1}{4}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{2}{3 \cdot 4} x^{\frac{3}{2}} + C_{2}$

Step 2.3.5.2.2

Multiply $3$ by $4$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{2}{12} x^{\frac{3}{2}} + C_{2}$

Step 2.3.5.2.3

Cancel the common factor of $2$ and $12$ .

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Step 2.3.5.2.3.1

Factor $2$ out of $2$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{2 (1)}{12} x^{\frac{3}{2}} + C_{2}$

Step 2.3.5.2.3.2

Cancel the common factors.

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Step 2.3.5.2.3.2.1

Factor $2$ out of $12$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{2 \cdot 1}{2 \cdot 6} x^{\frac{3}{2}} + C_{2}$

Step 2.3.5.2.3.2.2

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = - \frac{2 \cdot 1}{2 \cdot 6} x^{\frac{3}{2}} + C_{2}$

Step 2.3.5.2.3.2.3

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{6} x^{\frac{3}{2}} + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = - \frac{1}{6} x^{\frac{3}{2}} + K$