Calculus Examples

$\frac{d y}{d x} = {(64 x y)}^{\frac{1}{3}}$

Step 1

Separate the variables.

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Step 1.1

Factor.

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Step 1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.1.1

Apply the product rule to $64 x y$ .

$\frac{d y}{d x} = {(64 x)}^{\frac{1}{3}} y^{\frac{1}{3}}$

Step 1.1.1.2

Apply the product rule to $64 x$ .

$\frac{d y}{d x} = 64^{\frac{1}{3}} x^{\frac{1}{3}} y^{\frac{1}{3}}$

Step 1.1.2

Rewrite $64$ as $4^{3}$ .

$\frac{d y}{d x} = {(4^{3})}^{\frac{1}{3}} x^{\frac{1}{3}} y^{\frac{1}{3}}$

Step 1.1.3

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

$\frac{d y}{d x} = 4^{3 (\frac{1}{3})} x^{\frac{1}{3}} y^{\frac{1}{3}}$

Step 1.1.4

Cancel the common factor of $3$ .

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Step 1.1.4.1

Cancel the common factor.

$\frac{d y}{d x} = 4^{3 (\frac{1}{3})} x^{\frac{1}{3}} y^{\frac{1}{3}}$

Step 1.1.4.2

Rewrite the expression.

$\frac{d y}{d x} = 4^{1} x^{\frac{1}{3}} y^{\frac{1}{3}}$

Step 1.1.5

Evaluate the exponent.

$\frac{d y}{d x} = 4 x^{\frac{1}{3}} y^{\frac{1}{3}}$

Step 1.2

Multiply both sides by $\frac{1}{y^{\frac{1}{3}}}$ .

$\frac{1}{y^{\frac{1}{3}}} \frac{d y}{d x} = \frac{1}{y^{\frac{1}{3}}} (4 x^{\frac{1}{3}} y^{\frac{1}{3}})$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{\frac{1}{3}}} \frac{d y}{d x} = 4 \frac{1}{y^{\frac{1}{3}}} (x^{\frac{1}{3}} y^{\frac{1}{3}})$

Step 1.3.2

Combine $4$ and $\frac{1}{y^{\frac{1}{3}}}$ .

$\frac{1}{y^{\frac{1}{3}}} \frac{d y}{d x} = \frac{4}{y^{\frac{1}{3}}} (x^{\frac{1}{3}} y^{\frac{1}{3}})$

Step 1.3.3

Cancel the common factor of $y^{\frac{1}{3}}$ .

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Step 1.3.3.1

Factor $y^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} y^{\frac{1}{3}}$ .

$\frac{1}{y^{\frac{1}{3}}} \frac{d y}{d x} = \frac{4}{y^{\frac{1}{3}}} (y^{\frac{1}{3}} x^{\frac{1}{3}})$

Step 1.3.3.2

Cancel the common factor.

$\frac{1}{y^{\frac{1}{3}}} \frac{d y}{d x} = \frac{4}{y^{\frac{1}{3}}} (y^{\frac{1}{3}} x^{\frac{1}{3}})$

Step 1.3.3.3

Rewrite the expression.

$\frac{1}{y^{\frac{1}{3}}} \frac{d y}{d x} = 4 x^{\frac{1}{3}}$

Step 1.4

Rewrite the equation.

$\frac{1}{y^{\frac{1}{3}}} d y = 4 x^{\frac{1}{3}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{\frac{1}{3}}} d y = \int 4 x^{\frac{1}{3}} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Move $y^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{\frac{1}{3}})}^{- 1} d y = \int 4 x^{\frac{1}{3}} d x$

Step 2.2.1.2

Multiply the exponents in ${(y^{\frac{1}{3}})}^{- 1}$ .

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Step 2.2.1.2.1

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

$\int y^{\frac{1}{3} \cdot - 1} d y = \int 4 x^{\frac{1}{3}} d x$

Step 2.2.1.2.2

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

$\int y^{\frac{- 1}{3}} d y = \int 4 x^{\frac{1}{3}} d x$

Step 2.2.1.2.3

Move the negative in front of the fraction.

$\int y^{- \frac{1}{3}} d y = \int 4 x^{\frac{1}{3}} d x$

Step 2.2.2

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

$\frac{3}{2} y^{\frac{2}{3}} + C_{1} = \int 4 x^{\frac{1}{3}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{3}{2} y^{\frac{2}{3}} + C_{1} = 4 \int x^{\frac{1}{3}} d x$

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $4$ and $\frac{3}{4}$ .

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

Step 2.3.3.2.2

Multiply $4$ by $3$ .

$\frac{3}{2} y^{\frac{2}{3}} + C_{1} = \frac{12}{4} x^{\frac{4}{3}} + C_{2}$

Step 2.3.3.2.3

Cancel the common factor of $12$ and $4$ .

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Step 2.3.3.2.3.1

Factor $4$ out of $12$ .

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

Step 2.3.3.2.3.2

Cancel the common factors.

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Step 2.3.3.2.3.2.1

Factor $4$ out of $4$ .

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

Step 2.3.3.2.3.2.2

Cancel the common factor.

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

Step 2.3.3.2.3.2.3

Rewrite the expression.

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

Step 2.3.3.2.3.2.4

Divide $3$ by $1$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} (\frac{3}{2} y^{\frac{2}{3}}) = \frac{2}{3} (3 x^{\frac{4}{3}} + K)$

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 $\frac{2}{3} (\frac{3}{2} y^{\frac{2}{3}})$ .

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Step 3.2.1.1.1

Combine $\frac{3}{2}$ and $y^{\frac{2}{3}}$ .

$\frac{2}{3} \cdot \frac{3 y^{\frac{2}{3}}}{2} = \frac{2}{3} (3 x^{\frac{4}{3}} + K)$

Step 3.2.1.1.2

Combine.

$\frac{2 (3 y^{\frac{2}{3}})}{3 \cdot 2} = \frac{2}{3} (3 x^{\frac{4}{3}} + K)$

Step 3.2.1.1.3

Cancel the common factor.

$\frac{2 (3 y^{\frac{2}{3}})}{3 \cdot 2} = \frac{2}{3} (3 x^{\frac{4}{3}} + K)$

Step 3.2.1.1.4

Rewrite the expression.

$\frac{3 y^{\frac{2}{3}}}{3} = \frac{2}{3} (3 x^{\frac{4}{3}} + K)$

Step 3.2.1.1.5

Cancel the common factor.

$\frac{3 y^{\frac{2}{3}}}{3} = \frac{2}{3} (3 x^{\frac{4}{3}} + K)$

Step 3.2.1.1.6

Divide $y^{\frac{2}{3}}$ by $1$ .

$y^{\frac{2}{3}} = \frac{2}{3} (3 x^{\frac{4}{3}} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{2}{3} (3 x^{\frac{4}{3}} + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$y^{\frac{2}{3}} = \frac{2}{3} (3 x^{\frac{4}{3}}) + \frac{2}{3} K$

Step 3.2.2.1.2

Cancel the common factor of $3$ .

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Step 3.2.2.1.2.1

Factor $3$ out of $3 x^{\frac{4}{3}}$ .

$y^{\frac{2}{3}} = \frac{2}{3} (3 (x^{\frac{4}{3}})) + \frac{2}{3} K$

Step 3.2.2.1.2.2

Cancel the common factor.

$y^{\frac{2}{3}} = \frac{2}{3} (3 x^{\frac{4}{3}}) + \frac{2}{3} K$

Step 3.2.2.1.2.3

Rewrite the expression.

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

Step 3.2.2.1.3

Combine $\frac{2}{3}$ and $K$ .

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

Step 3.3

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

Step 3.4

Simplify the left side.

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Step 3.4.1

Simplify ${(y^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 3.4.1.1

Multiply the exponents in ${(y^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 3.4.1.1.1

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

$y^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

Step 3.4.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.1.1.2.1

Cancel the common factor.

$y^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

Step 3.4.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{3} \cdot 3} = \pm {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

Step 3.4.1.1.3

Cancel the common factor of $3$ .

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Step 3.4.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = \pm {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

Step 3.4.1.1.3.2

Rewrite the expression.

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

Step 3.4.1.2

Simplify.

$y = \pm {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

Step 3.5

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

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Step 3.5.1

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

$y = {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

Step 3.5.2

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

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

Step 3.5.3

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

$y = {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

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

$y = {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

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

$y = {(2 x^{\frac{4}{3}} + \frac{2 K}{3})}^{\frac{3}{2}}$

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

Step 4

Simplify the constant of integration.

$y = {(2 x^{\frac{4}{3}} + K)}^{\frac{3}{2}}$

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