Calculus Examples

$\frac{d y}{d x} = \frac{8 x^{2}}{9 y^{2}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} \frac{8 x^{2}}{9 y^{2}}$

Step 1.2

Cancel the common factor of $y^{2}$ .

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Step 1.2.1

Factor $y^{2}$ out of $9 y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} \frac{8 x^{2}}{y^{2} \cdot 9}$

Step 1.2.2

Cancel the common factor.

$y^{2} \frac{d y}{d x} = y^{2} \frac{8 x^{2}}{y^{2} \cdot 9}$

Step 1.2.3

Rewrite the expression.

$y^{2} \frac{d y}{d x} = \frac{8 x^{2}}{9}$

Step 1.3

Rewrite the equation.

$y^{2} d y = \frac{8 x^{2}}{9} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} d y = \int \frac{8 x^{2}}{9} d x$

Step 2.2

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

$\frac{1}{3} y^{3} + C_{1} = \int \frac{8 x^{2}}{9} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{8}{9}$ is constant with respect to $x$ , move $\frac{8}{9}$ out of the integral.

$\frac{1}{3} y^{3} + C_{1} = \frac{8}{9} \int x^{2} d x$

Step 2.3.2

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

$\frac{1}{3} y^{3} + C_{1} = \frac{8}{9} (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $\frac{8}{9} (\frac{1}{3} x^{3} + C_{2})$ as $\frac{8}{9} \cdot \frac{1}{3} x^{3} + C_{2}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{8}{9} \cdot \frac{1}{3} x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Multiply $\frac{8}{9}$ by $\frac{1}{3}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{8}{9 \cdot 3} x^{3} + C_{2}$

Step 2.3.3.2.2

Multiply $9$ by $3$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{8}{27} x^{3} + C_{2}$

Step 2.4

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

$\frac{1}{3} y^{3} = \frac{8}{27} x^{3} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (\frac{8}{27} x^{3} + K)$

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 $3 (\frac{1}{3} y^{3})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{3}$ and $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (\frac{8}{27} x^{3} + K)$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$3 \frac{y^{3}}{3} = 3 (\frac{8}{27} x^{3} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (\frac{8}{27} x^{3} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (\frac{8}{27} x^{3} + K)$ .

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Step 3.2.2.1.1

Combine $\frac{8}{27}$ and $x^{3}$ .

$y^{3} = 3 (\frac{8 x^{3}}{27} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{3} = 3 \frac{8 x^{3}}{27} + 3 K$

Step 3.2.2.1.3

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.1

Factor $3$ out of $27$ .

$y^{3} = 3 \frac{8 x^{3}}{3 (9)} + 3 K$

Step 3.2.2.1.3.2

Cancel the common factor.

$y^{3} = 3 \frac{8 x^{3}}{3 \cdot 9} + 3 K$

Step 3.2.2.1.3.3

Rewrite the expression.

$y^{3} = \frac{8 x^{3}}{9} + 3 K$

Step 3.3

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

$y = \sqrt[3]{\frac{8 x^{3}}{9} + 3 K}$

Step 3.4

Simplify $\sqrt[3]{\frac{8 x^{3}}{9} + 3 K}$ .

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Step 3.4.1

To write $3 K$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = \sqrt[3]{\frac{8 x^{3}}{9} + 3 K \cdot \frac{9}{9}}$

Step 3.4.2

Combine $3 K$ and $\frac{9}{9}$ .

$y = \sqrt[3]{\frac{8 x^{3}}{9} + \frac{3 K \cdot 9}{9}}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{8 x^{3} + 3 K \cdot 9}{9}}$

Step 3.4.4

Multiply $9$ by $3$ .

$y = \sqrt[3]{\frac{8 x^{3} + 27 K}{9}}$

Step 3.4.5

Rewrite $\sqrt[3]{\frac{8 x^{3} + 27 K}{9}}$ as $\frac{\sqrt[3]{8 x^{3} + 27 K}}{\sqrt[3]{9}}$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K}}{\sqrt[3]{9}}$

Step 3.4.6

Multiply $\frac{\sqrt[3]{8 x^{3} + 27 K}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K}}{\sqrt[3]{9}} \cdot \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$

Step 3.4.7

Combine and simplify the denominator.

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Step 3.4.7.1

Multiply $\frac{\sqrt[3]{8 x^{3} + 27 K}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{\sqrt[3]{9} {\sqrt[3]{9}}^{2}}$

Step 3.4.7.2

Raise $\sqrt[3]{9}$ to the power of $1$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{2}}$

Step 3.4.7.3

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

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1 + 2}}$

Step 3.4.7.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{3}}$

Step 3.4.7.5

Rewrite ${\sqrt[3]{9}}^{3}$ as $9$ .

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Step 3.4.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{{(9^{\frac{1}{3}})}^{3}}$

Step 3.4.7.5.2

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

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{9^{\frac{1}{3} \cdot 3}}$

Step 3.4.7.5.3

Combine $\frac{1}{3}$ and $3$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 3.4.7.5.4

Cancel the common factor of $3$ .

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Step 3.4.7.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 3.4.7.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{9^{1}}$

Step 3.4.7.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} {\sqrt[3]{9}}^{2}}{9}$

Step 3.4.8

Simplify the numerator.

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Step 3.4.8.1

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} \sqrt[3]{9^{2}}}{9}$

Step 3.4.8.2

Raise $9$ to the power of $2$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} \sqrt[3]{81}}{9}$

Step 3.4.8.3

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 3.4.8.3.1

Factor $27$ out of $81$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} \sqrt[3]{27 (3)}}{9}$

Step 3.4.8.3.2

Rewrite $27$ as $3^{3}$ .

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} \sqrt[3]{3^{3} \cdot 3}}{9}$

Step 3.4.8.4

Pull terms out from under the radical.

$y = \frac{\sqrt[3]{8 x^{3} + 27 K} \cdot 3 \sqrt[3]{3}}{9}$

Step 3.4.8.5

Combine using the product rule for radicals.

$y = \frac{3 \sqrt[3]{(8 x^{3} + 27 K) \cdot 3}}{9}$

Step 3.4.9

Reduce the expression by cancelling the common factors.

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Step 3.4.9.1

Cancel the common factor of $3$ and $9$ .

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Step 3.4.9.1.1

Factor $3$ out of $3 \sqrt[3]{(8 x^{3} + 27 K) \cdot 3}$ .

$y = \frac{3 \sqrt[3]{(8 x^{3} + 27 K) \cdot 3}}{9}$

Step 3.4.9.1.2

Cancel the common factors.

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Step 3.4.9.1.2.1

Factor $3$ out of $9$ .

$y = \frac{3 \sqrt[3]{(8 x^{3} + 27 K) \cdot 3}}{3 \cdot 3}$

Step 3.4.9.1.2.2

Cancel the common factor.

$y = \frac{3 \sqrt[3]{(8 x^{3} + 27 K) \cdot 3}}{3 \cdot 3}$

Step 3.4.9.1.2.3

Rewrite the expression.

$y = \frac{\sqrt[3]{(8 x^{3} + 27 K) \cdot 3}}{3}$

Step 3.4.9.2

Reorder factors in $\frac{\sqrt[3]{(8 x^{3} + 27 K) \cdot 3}}{3}$ .

$y = \frac{\sqrt[3]{3 (8 x^{3} + 27 K)}}{3}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{3 (8 x^{3} + K)}}{3}$