Calculus Examples

$\frac{d y}{d x} = \frac{3 x^{2}}{y^{4}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y^{4}$ .

$y^{4} \frac{d y}{d x} = y^{4} \frac{3 x^{2}}{y^{4}}$

Step 1.2

Cancel the common factor of $y^{4}$ .

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Step 1.2.1

Cancel the common factor.

$y^{4} \frac{d y}{d x} = y^{4} \frac{3 x^{2}}{y^{4}}$

Step 1.2.2

Rewrite the expression.

$y^{4} \frac{d y}{d x} = 3 x^{2}$

Step 1.3

Rewrite the equation.

$y^{4} d y = 3 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{4} d y = \int 3 x^{2} d x$

Step 2.2

By the Power Rule, the integral of $y^{4}$ with respect to $y$ is $\frac{1}{5} y^{5}$ .

$\frac{1}{5} y^{5} + C_{1} = \int 3 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$\frac{1}{5} y^{5} + C_{1} = 3 \int x^{2} d x$

Step 2.3.2

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

$\frac{1}{5} y^{5} + C_{1} = 3 (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

$\frac{1}{5} y^{5} + C_{1} = 3 (\frac{1}{3}) x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

$\frac{1}{5} y^{5} + C_{1} = \frac{3}{3} x^{3} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $3$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

$\frac{1}{5} y^{5} + C_{1} = \frac{3}{3} x^{3} + C_{2}$

Step 2.3.3.2.2.2

Rewrite the expression.

$\frac{1}{5} y^{5} + C_{1} = 1 x^{3} + C_{2}$

Step 2.3.3.2.3

Multiply $x^{3}$ by $1$ .

$\frac{1}{5} y^{5} + C_{1} = x^{3} + C_{2}$

Step 2.4

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

$\frac{1}{5} y^{5} = x^{3} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $5$ .

$5 (\frac{1}{5} y^{5}) = 5 (x^{3} + K)$

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 $5 (\frac{1}{5} y^{5})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{5}$ and $y^{5}$ .

$5 \frac{y^{5}}{5} = 5 (x^{3} + K)$

Step 3.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$5 \frac{y^{5}}{5} = 5 (x^{3} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{5} = 5 (x^{3} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$y^{5} = 5 x^{3} + 5 K$

Step 3.3

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

$y = \sqrt[5]{5 x^{3} + 5 K}$

Step 3.4

Factor $5$ out of $5 x^{3} + 5 K$ .

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Step 3.4.1

Factor $5$ out of $5 x^{3}$ .

$y = \sqrt[5]{5 (x^{3}) + 5 K}$

Step 3.4.2

Factor $5$ out of $5 (x^{3}) + 5 K$ .

$y = \sqrt[5]{5 (x^{3} + K)}$