Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y^{\frac{2}{3}}}$ .

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

Step 1.2

Cancel the common factor of $y^{\frac{2}{3}}$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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 $y^{\frac{2}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{\frac{2}{3}})}^{- 1} d y = \int d x$

Step 2.2.1.2

Multiply the exponents in ${(y^{\frac{2}{3}})}^{- 1}$ .

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Step 2.2.1.2.1

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

$\int y^{\frac{2}{3} \cdot - 1} d y = \int d x$

Step 2.2.1.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 2.2.1.2.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$\int y^{\frac{2 \cdot - 1}{3}} d y = \int d x$

Step 2.2.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.1.2.3

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $3 y^{\frac{1}{3}} = x + K$ by $3$ and simplify.

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Step 3.1.1

Divide each term in $3 y^{\frac{1}{3}} = x + K$ by $3$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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

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

Step 3.2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Step 3.3

Simplify the exponent.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

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Step 3.3.1.1.1.1

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

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

Step 3.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.2.1

Cancel the common factor.

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

Step 3.3.1.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.2

Simplify.

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Use the Binomial Theorem.

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

Step 3.3.2.1.2

Simplify each term.

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Step 3.3.2.1.2.1

Apply the product rule to $\frac{x}{3}$ .

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

Step 3.3.2.1.2.2

Raise $3$ to the power of $3$ .

$y = \frac{x^{3}}{27} + 3 {(\frac{x}{3})}^{2} \frac{K}{3} + 3 \frac{x}{3} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.3

Cancel the common factor of $3$ .

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Step 3.3.2.1.2.3.1

Factor $3$ out of $3 {(\frac{x}{3})}^{2}$ .

$y = \frac{x^{3}}{27} + 3 ({(\frac{x}{3})}^{2}) \frac{K}{3} + 3 \frac{x}{3} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.3.2

Cancel the common factor.

$y = \frac{x^{3}}{27} + 3 {(\frac{x}{3})}^{2} \frac{K}{3} + 3 \frac{x}{3} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.3.3

Rewrite the expression.

$y = \frac{x^{3}}{27} + {(\frac{x}{3})}^{2} K + 3 \frac{x}{3} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.4

Apply the product rule to $\frac{x}{3}$ .

$y = \frac{x^{3}}{27} + \frac{x^{2}}{3^{2}} K + 3 \frac{x}{3} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.5

Raise $3$ to the power of $2$ .

$y = \frac{x^{3}}{27} + \frac{x^{2}}{9} K + 3 \frac{x}{3} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.6

Combine $\frac{x^{2}}{9}$ and $K$ .

$y = \frac{x^{3}}{27} + \frac{x^{2} K}{9} + 3 \frac{x}{3} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.7

Cancel the common factor of $3$ .

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Step 3.3.2.1.2.7.1

Cancel the common factor.

$y = \frac{x^{3}}{27} + \frac{x^{2} K}{9} + 3 \frac{x}{3} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.7.2

Rewrite the expression.

$y = \frac{x^{3}}{27} + \frac{x^{2} K}{9} + x {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.8

Apply the product rule to $\frac{K}{3}$ .

$y = \frac{x^{3}}{27} + \frac{x^{2} K}{9} + x \frac{K^{2}}{3^{2}} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.9

Raise $3$ to the power of $2$ .

$y = \frac{x^{3}}{27} + \frac{x^{2} K}{9} + x \frac{K^{2}}{9} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.10

Combine $x$ and $\frac{K^{2}}{9}$ .

$y = \frac{x^{3}}{27} + \frac{x^{2} K}{9} + \frac{x K^{2}}{9} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.11

Apply the product rule to $\frac{K}{3}$ .

$y = \frac{x^{3}}{27} + \frac{x^{2} K}{9} + \frac{x K^{2}}{9} + \frac{K^{3}}{3^{3}}$

Step 3.3.2.1.2.12

Raise $3$ to the power of $3$ .

$y = \frac{x^{3}}{27} + \frac{x^{2} K}{9} + \frac{x K^{2}}{9} + \frac{K^{3}}{27}$

Step 3.4

Simplify $\frac{x^{3}}{27} + \frac{x^{2} K}{9} + \frac{x K^{2}}{9} + \frac{K^{3}}{27}$ .

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Step 3.4.1

Reorder $x^{2}$ and $K$ .

$y = \frac{x^{3}}{27} + \frac{K x^{2}}{9} + \frac{x K^{2}}{9} + \frac{K^{3}}{27}$

Step 3.4.2

Reorder $x$ and $K^{2}$ .

$y = \frac{x^{3}}{27} + \frac{K x^{2}}{9} + \frac{K^{2} x}{9} + \frac{K^{3}}{27}$

Step 3.4.3

Move $\frac{x^{3}}{27}$ .

$y = \frac{K x^{2}}{9} + \frac{K^{2} x}{9} + \frac{K^{3}}{27} + \frac{x^{3}}{27}$

Step 3.4.4

Move $\frac{K x^{2}}{9}$ .

$y = \frac{K^{2} x}{9} + \frac{K^{3}}{27} + \frac{K x^{2}}{9} + \frac{x^{3}}{27}$

Step 3.4.5

Reorder $\frac{K^{2} x}{9}$ and $\frac{K^{3}}{27}$ .

$y = \frac{K^{3}}{27} + \frac{K^{2} x}{9} + \frac{K x^{2}}{9} + \frac{x^{3}}{27}$

Step 4

Simplify the constant of integration.

$y = K + \frac{K x}{9} + \frac{K x^{2}}{9} + \frac{x^{3}}{27}$