Calculus Examples

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

Step 1

Write the problem as a mathematical expression.

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

Step 2

Separate the variables.

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Step 2.1

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

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Step 2.1.1

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

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

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.

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

Step 2.1.2.1.2

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

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

Step 2.2

Multiply both sides by $y$ .

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

Step 2.3

Cancel the common factor of $y$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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

Step 2.4

Rewrite the equation.

$y d y = x^{3} d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int y d y = \int x^{3} d x$

Step 3.2

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

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

Step 3.3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 3.4

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

$\frac{1}{2} y^{2} = \frac{1}{4} x^{4} + K$

Step 4

Solve for $y$ .

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Step 4.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{1}{4} x^{4} + K)$

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 $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (\frac{1}{4} x^{4} + K)$

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.

$2 \frac{y^{2}}{2} = 2 (\frac{1}{4} x^{4} + K)$

Step 4.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{1}{4} x^{4} + K)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $2 (\frac{1}{4} x^{4} + K)$ .

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Step 4.2.2.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$y^{2} = 2 (\frac{x^{4}}{4} + K)$

Step 4.2.2.1.2

Apply the distributive property.

$y^{2} = 2 \frac{x^{4}}{4} + 2 K$

Step 4.2.2.1.3

Cancel the common factor of $2$ .

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Step 4.2.2.1.3.1

Factor $2$ out of $4$ .

$y^{2} = 2 \frac{x^{4}}{2 (2)} + 2 K$

Step 4.2.2.1.3.2

Cancel the common factor.

$y^{2} = 2 \frac{x^{4}}{2 \cdot 2} + 2 K$

Step 4.2.2.1.3.3

Rewrite the expression.

$y^{2} = \frac{x^{4}}{2} + 2 K$

Step 4.3

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

$y = \pm \sqrt{\frac{x^{4}}{2} + 2 K}$

Step 4.4

Simplify $\pm \sqrt{\frac{x^{4}}{2} + 2 K}$ .

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Step 4.4.1

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

$y = \pm \sqrt{\frac{x^{4}}{2} + 2 K \cdot \frac{2}{2}}$

Step 4.4.2

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

$y = \pm \sqrt{\frac{x^{4}}{2} + \frac{2 K \cdot 2}{2}}$

Step 4.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{4} + 2 K \cdot 2}{2}}$

Step 4.4.4

Multiply $2$ by $2$ .

$y = \pm \sqrt{\frac{x^{4} + 4 K}{2}}$

Step 4.4.5

Rewrite $\sqrt{\frac{x^{4} + 4 K}{2}}$ as $\frac{\sqrt{x^{4} + 4 K}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x^{4} + 4 K}}{\sqrt{2}}$

Step 4.4.6

Multiply $\frac{\sqrt{x^{4} + 4 K}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x^{4} + 4 K}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.4.7

Combine and simplify the denominator.

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Step 4.4.7.1

Multiply $\frac{\sqrt{x^{4} + 4 K}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 4.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 4.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.4.7.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.4.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.4.7.6.1

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

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.4.7.6.2

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

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.4.7.6.3

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

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.4.7.6.4

Cancel the common factor of $2$ .

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Step 4.4.7.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.4.7.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{2^{1}}$

Step 4.4.7.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x^{4} + 4 K} \sqrt{2}}{2}$

Step 4.4.8

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x^{4} + 4 K) \cdot 2}}{2}$

Step 4.4.9

Reorder factors in $\pm \frac{\sqrt{(x^{4} + 4 K) \cdot 2}}{2}$ .

$y = \pm \frac{\sqrt{2 (x^{4} + 4 K)}}{2}$

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.

$y = \frac{\sqrt{2 (x^{4} + 4 K)}}{2}$

Step 4.5.2

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

$y = - \frac{\sqrt{2 (x^{4} + 4 K)}}{2}$

Step 4.5.3

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

$y = \frac{\sqrt{2 (x^{4} + 4 K)}}{2}$

$y = - \frac{\sqrt{2 (x^{4} + 4 K)}}{2}$

$y = \frac{\sqrt{2 (x^{4} + 4 K)}}{2}$

$y = - \frac{\sqrt{2 (x^{4} + 4 K)}}{2}$

$y = \frac{\sqrt{2 (x^{4} + 4 K)}}{2}$

$y = - \frac{\sqrt{2 (x^{4} + 4 K)}}{2}$

Step 5

Simplify the constant of integration.

$y = \frac{\sqrt{2 (x^{4} + K)}}{2}$

$y = - \frac{\sqrt{2 (x^{4} + K)}}{2}$