Calculus Examples

$\frac{d y}{d x} = 6 x {(y - 1)}^{\frac{2}{3}}$

Step 1

Separate the variables.

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Step 1.1

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

$\frac{1}{{(y - 1)}^{\frac{2}{3}}} \frac{d y}{d x} = \frac{1}{{(y - 1)}^{\frac{2}{3}}} (6 x {(y - 1)}^{\frac{2}{3}})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{{(y - 1)}^{\frac{2}{3}}} \frac{d y}{d x} = 6 \frac{1}{{(y - 1)}^{\frac{2}{3}}} (x {(y - 1)}^{\frac{2}{3}})$

Step 1.2.2

Combine $6$ and $\frac{1}{{(y - 1)}^{\frac{2}{3}}}$ .

$\frac{1}{{(y - 1)}^{\frac{2}{3}}} \frac{d y}{d x} = \frac{6}{{(y - 1)}^{\frac{2}{3}}} (x {(y - 1)}^{\frac{2}{3}})$

Step 1.2.3

Cancel the common factor of ${(y - 1)}^{\frac{2}{3}}$ .

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Step 1.2.3.1

Factor ${(y - 1)}^{\frac{2}{3}}$ out of $x {(y - 1)}^{\frac{2}{3}}$ .

$\frac{1}{{(y - 1)}^{\frac{2}{3}}} \frac{d y}{d x} = \frac{6}{{(y - 1)}^{\frac{2}{3}}} ({(y - 1)}^{\frac{2}{3}} x)$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{{(y - 1)}^{\frac{2}{3}}} \frac{d y}{d x} = \frac{6}{{(y - 1)}^{\frac{2}{3}}} ({(y - 1)}^{\frac{2}{3}} x)$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{{(y - 1)}^{\frac{2}{3}}} \frac{d y}{d x} = 6 x$

Step 1.3

Rewrite the equation.

$\frac{1}{{(y - 1)}^{\frac{2}{3}}} d y = 6 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{{(y - 1)}^{\frac{2}{3}}} d y = \int 6 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = y - 1$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = y - 1$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $y - 1$ .

$\frac{d}{d y} [y - 1]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $y - 1$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [- 1]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [- 1]$

Step 2.2.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d y} [- 1]$

Step 2.2.1.1.4

Since $- 1$ is constant with respect to $y$ , the derivative of $- 1$ with respect to $y$ is $0$ .

$1 + 0$

Step 2.2.1.1.5

Add $1$ and $0$ .

$1$

Step 2.2.1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u^{\frac{2}{3}}} d u = \int 6 x d x$

Step 2.2.2

Apply basic rules of exponents.

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Step 2.2.2.1

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

$\int {(u^{\frac{2}{3}})}^{- 1} d u = \int 6 x d x$

Step 2.2.2.2

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

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Step 2.2.2.2.1

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

$\int u^{\frac{2}{3} \cdot - 1} d u = \int 6 x d x$

Step 2.2.2.2.2

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

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Step 2.2.2.2.2.1

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

$\int u^{\frac{2 \cdot - 1}{3}} d u = \int 6 x d x$

Step 2.2.2.2.2.2

Multiply $2$ by $- 1$ .

$\int u^{\frac{- 2}{3}} d u = \int 6 x d x$

Step 2.2.2.2.3

Move the negative in front of the fraction.

$\int u^{- \frac{2}{3}} d u = \int 6 x d x$

Step 2.2.3

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

$3 u^{\frac{1}{3}} + C_{1} = \int 6 x d x$

Step 2.2.4

Replace all occurrences of $u$ with $y - 1$ .

$3 {(y - 1)}^{\frac{1}{3}} + C_{1} = \int 6 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$3 {(y - 1)}^{\frac{1}{3}} + C_{1} = 6 \int x d x$

Step 2.3.2

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

$3 {(y - 1)}^{\frac{1}{3}} + C_{1} = 6 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $6 (\frac{1}{2} x^{2} + C_{2})$ as $6 (\frac{1}{2}) x^{2} + C_{2}$ .

$3 {(y - 1)}^{\frac{1}{3}} + C_{1} = 6 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $6$ and $\frac{1}{2}$ .

$3 {(y - 1)}^{\frac{1}{3}} + C_{1} = \frac{6}{2} x^{2} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $6$ and $2$ .

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Step 2.3.3.2.2.1

Factor $2$ out of $6$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.3.2.2.2.4

Divide $3$ by $1$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $3 {(y - 1)}^{\frac{1}{3}} = 3 x^{2} + K$ by $3$ and simplify.

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Step 3.1.1

Divide each term in $3 {(y - 1)}^{\frac{1}{3}} = 3 x^{2} + K$ by $3$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Cancel the common factor of $3$ .

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Step 3.1.3.1.1

Cancel the common factor.

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

Step 3.1.3.1.2

Divide $x^{2}$ by $1$ .

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

Step 3.2

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

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

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

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Step 3.3.1.1.1

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

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Step 3.3.1.1.1.1

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

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

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.

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

Step 3.3.1.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.2

Simplify.

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Use the Binomial Theorem.

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

Step 3.3.2.1.2

Simplify each term.

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Step 3.3.2.1.2.1

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 3.3.2.1.2.1.1

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

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

Step 3.3.2.1.2.1.2

Multiply $2$ by $3$ .

$y - 1 = x^{6} + 3 {(x^{2})}^{2} \frac{K}{3} + 3 x^{2} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.2

Cancel the common factor of $3$ .

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Step 3.3.2.1.2.2.1

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

$y - 1 = x^{6} + 3 ({(x^{2})}^{2}) \frac{K}{3} + 3 x^{2} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.2.2

Cancel the common factor.

$y - 1 = x^{6} + 3 {(x^{2})}^{2} \frac{K}{3} + 3 x^{2} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.2.3

Rewrite the expression.

$y - 1 = x^{6} + {(x^{2})}^{2} K + 3 x^{2} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.3

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.3.2.1.2.3.1

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

$y - 1 = x^{6} + x^{2 \cdot 2} K + 3 x^{2} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.3.2

Multiply $2$ by $2$ .

$y - 1 = x^{6} + x^{4} K + 3 x^{2} {(\frac{K}{3})}^{2} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.4

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

$y - 1 = x^{6} + x^{4} K + 3 x^{2} \frac{K^{2}}{3^{2}} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.5

Cancel the common factor of $3$ .

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Step 3.3.2.1.2.5.1

Factor $3$ out of $3 x^{2}$ .

$y - 1 = x^{6} + x^{4} K + 3 (x^{2}) \frac{K^{2}}{3^{2}} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.5.2

Factor $3$ out of $3^{2}$ .

$y - 1 = x^{6} + x^{4} K + 3 (x^{2}) \frac{K^{2}}{3 \cdot 3} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.5.3

Cancel the common factor.

$y - 1 = x^{6} + x^{4} K + 3 x^{2} \frac{K^{2}}{3 \cdot 3} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.5.4

Rewrite the expression.

$y - 1 = x^{6} + x^{4} K + x^{2} \frac{K^{2}}{3} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.6

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

$y - 1 = x^{6} + x^{4} K + \frac{x^{2} K^{2}}{3} + {(\frac{K}{3})}^{3}$

Step 3.3.2.1.2.7

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

$y - 1 = x^{6} + x^{4} K + \frac{x^{2} K^{2}}{3} + \frac{K^{3}}{3^{3}}$

Step 3.3.2.1.2.8

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

$y - 1 = x^{6} + x^{4} K + \frac{x^{2} K^{2}}{3} + \frac{K^{3}}{27}$

Step 3.4

Add $1$ to both sides of the equation.

$y = x^{6} + x^{4} K + \frac{x^{2} K^{2}}{3} + \frac{K^{3}}{27} + 1$

Step 4

Simplify the constant of integration.

$y = x^{6} + x^{4} K + \frac{x^{2} K}{3} + K$