Calculus Examples

$\frac{d y}{d x}$ for $y = π^{3}$

Step 1

Write the problem as a mathematical expression.

$\frac{d y}{d x} \cdot y = π^{3}$

Step 2

Separate the variables.

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Step 2.1

Divide each term in $\frac{d y}{d x} \cdot y = π^{3}$ by $y$ and simplify.

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Step 2.1.1

Divide each term in $\frac{d y}{d x} \cdot y = π^{3}$ by $y$ .

$\frac{\frac{d y}{d x} \cdot y}{y} = \frac{π^{3}}{y}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $y$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{\frac{d y}{d x} \cdot y}{y} = \frac{π^{3}}{y}$

Step 2.1.2.1.2

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

$\frac{d y}{d x} = \frac{π^{3}}{y}$

Step 2.2

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{π^{3}}{y}$

Step 2.3

Cancel the common factor of $y$ .

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Step 2.3.1

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{π^{3}}{y}$

Step 2.3.2

Rewrite the expression.

$y \frac{d y}{d x} = π^{3}$

Step 2.4

Rewrite the equation.

$y d y = π^{3} d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int y d y = \int π^{3} d x$

Step 3.2

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

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

Step 3.3

Apply the constant rule.

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

Step 3.4

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

$\frac{1}{2} y^{2} = π^{3} x + K$

Step 4

Solve for $y$ .

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Step 4.1

Multiply both sides of the equation by $2$ .

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

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (π^{3} x + K)$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (π^{3} x + K)$

Step 4.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (π^{3} x + K)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Apply the distributive property.

$y^{2} = 2 π^{3} x + 2 K$

Step 4.3

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

$y = \pm \sqrt{2 π^{3} x + 2 K}$

Step 4.4

Factor $2$ out of $2 π^{3} x + 2 K$ .

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Step 4.4.1

Factor $2$ out of $2 π^{3} x$ .

$y = \pm \sqrt{2 (π^{3} x) + 2 K}$

Step 4.4.2

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (π^{3} x) + 2 K}$

Step 4.4.3

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

$y = \pm \sqrt{2 (π^{3} x + K)}$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{2 (π^{3} x + K)}$

Step 4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{2 (π^{3} x + K)}$

Step 4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{2 (π^{3} x + K)}$

$y = - \sqrt{2 (π^{3} x + K)}$

$y = \sqrt{2 (π^{3} x + K)}$

$y = - \sqrt{2 (π^{3} x + K)}$

$y = \sqrt{2 (π^{3} x + K)}$

$y = - \sqrt{2 (π^{3} x + K)}$