Calculus Examples

$\frac{d y}{d t} = y^{4}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y^{4}}$ .

$\frac{1}{y^{4}} \frac{d y}{d t} = \frac{1}{y^{4}} y^{4}$

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

$\frac{1}{y^{4}} \frac{d y}{d t} = \frac{1}{y^{4}} y^{4}$

Step 1.2.2

Rewrite the expression.

$\frac{1}{y^{4}} \frac{d y}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{4}} d y = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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 $y^{4}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{4})}^{- 1} d y = \int d t$

Step 2.2.1.2

Multiply the exponents in ${(y^{4})}^{- 1}$ .

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Step 2.2.1.2.1

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

$\int y^{4 \cdot - 1} d y = \int d t$

Step 2.2.1.2.2

Multiply $4$ by $- 1$ .

$\int y^{- 4} d y = \int d t$

Step 2.2.2

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

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $- \frac{1}{3} y^{- 3} + C_{1}$ as $- \frac{1}{3} \cdot \frac{1}{y^{3}} + C_{1}$ .

$- \frac{1}{3} \cdot \frac{1}{y^{3}} + C_{1} = \int d t$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

$- \frac{1}{y^{3} \cdot 3} + C_{1} = \int d t$

Step 2.2.3.2.2

Move $3$ to the left of $y^{3}$ .

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 y^{3}, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$3 y^{3}$

Step 3.2

Multiply each term in $- \frac{1}{3 y^{3}} = t + K$ by $3 y^{3}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $- \frac{1}{3 y^{3}} = t + K$ by $3 y^{3}$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Move the leading negative in $- \frac{1}{3 y^{3}}$ into the numerator.

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

Step 3.2.2.1.2

Cancel the common factor.

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

Step 3.2.2.1.3

Rewrite the expression.

$- 1 = t (3 y^{3}) + K (3 y^{3})$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Rewrite using the commutative property of multiplication.

$- 1 = 3 t y^{3} + K (3 y^{3})$

Step 3.2.3.1.2

Rewrite using the commutative property of multiplication.

$- 1 = 3 t y^{3} + 3 K y^{3}$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $3 t y^{3} + 3 K y^{3} = - 1$ .

$3 t y^{3} + 3 K y^{3} = - 1$

Step 3.3.2

Factor $3 y^{3}$ out of $3 t y^{3} + 3 K y^{3}$ .

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Step 3.3.2.1

Factor $3 y^{3}$ out of $3 t y^{3}$ .

$3 y^{3} (t) + 3 K y^{3} = - 1$

Step 3.3.2.2

Factor $3 y^{3}$ out of $3 K y^{3}$ .

$3 y^{3} (t) + 3 y^{3} K = - 1$

Step 3.3.2.3

Factor $3 y^{3}$ out of $3 y^{3} (t) + 3 y^{3} K$ .

$3 y^{3} (t + K) = - 1$

Step 3.3.3

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

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Step 3.3.3.1

Divide each term in $3 y^{3} (t + K) = - 1$ by $3 (t + K)$ .

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

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.1.2

Rewrite the expression.

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

Step 3.3.3.2.2

Cancel the common factor of $t + K$ .

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Step 3.3.3.2.2.1

Cancel the common factor.

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

Step 3.3.3.2.2.2

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

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

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

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

Step 3.3.4

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

$y = \sqrt[3]{- \frac{1}{3 (t + K)}}$

Step 3.3.5

Simplify $\sqrt[3]{- \frac{1}{3 (t + K)}}$ .

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Step 3.3.5.1

Rewrite $- \frac{1}{3 (t + K)}$ as ${({(- 1)}^{3})}^{3} \frac{1}{3 (t + K)}$ .

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Step 3.3.5.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \sqrt[3]{{(- 1)}^{3} \frac{1}{3 (t + K)}}$

Step 3.3.5.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{1}{3 (t + K)}}$

Step 3.3.5.2

Pull terms out from under the radical.

$y = {(- 1)}^{3} \sqrt[3]{\frac{1}{3 (t + K)}}$

Step 3.3.5.3

Raise $- 1$ to the power of $3$ .

$y = - \sqrt[3]{\frac{1}{3 (t + K)}}$

Step 3.3.5.4

Rewrite $\sqrt[3]{\frac{1}{3 (t + K)}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{3 (t + K)}}$ .

$y = - \frac{\sqrt[3]{1}}{\sqrt[3]{3 (t + K)}}$

Step 3.3.5.5

Any root of $1$ is $1$ .

$y = - \frac{1}{\sqrt[3]{3 (t + K)}}$

Step 3.3.5.6

Multiply $\frac{1}{\sqrt[3]{3 (t + K)}}$ by $\frac{{\sqrt[3]{3 (t + K)}}^{2}}{{\sqrt[3]{3 (t + K)}}^{2}}$ .

$y = - (\frac{1}{\sqrt[3]{3 (t + K)}} \cdot \frac{{\sqrt[3]{3 (t + K)}}^{2}}{{\sqrt[3]{3 (t + K)}}^{2}})$

Step 3.3.5.7

Combine and simplify the denominator.

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Step 3.3.5.7.1

Multiply $\frac{1}{\sqrt[3]{3 (t + K)}}$ by $\frac{{\sqrt[3]{3 (t + K)}}^{2}}{{\sqrt[3]{3 (t + K)}}^{2}}$ .

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{\sqrt[3]{3 (t + K)} {\sqrt[3]{3 (t + K)}}^{2}}$

Step 3.3.5.7.2

Raise $\sqrt[3]{3 (t + K)}$ to the power of $1$ .

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{{\sqrt[3]{3 (t + K)}}^{1} {\sqrt[3]{3 (t + K)}}^{2}}$

Step 3.3.5.7.3

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

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{{\sqrt[3]{3 (t + K)}}^{1 + 2}}$

Step 3.3.5.7.4

Add $1$ and $2$ .

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{{\sqrt[3]{3 (t + K)}}^{3}}$

Step 3.3.5.7.5

Rewrite ${\sqrt[3]{3 (t + K)}}^{3}$ as $3 (t + K)$ .

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Step 3.3.5.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 (t + K)}$ as ${(3 (t + K))}^{\frac{1}{3}}$ .

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{{({(3 (t + K))}^{\frac{1}{3}})}^{3}}$

Step 3.3.5.7.5.2

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

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{{(3 (t + K))}^{\frac{1}{3} \cdot 3}}$

Step 3.3.5.7.5.3

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

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{{(3 (t + K))}^{\frac{3}{3}}}$

Step 3.3.5.7.5.4

Cancel the common factor of $3$ .

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Step 3.3.5.7.5.4.1

Cancel the common factor.

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{{(3 (t + K))}^{\frac{3}{3}}}$

Step 3.3.5.7.5.4.2

Rewrite the expression.

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{{(3 (t + K))}^{1}}$

Step 3.3.5.7.5.5

Simplify.

$y = - \frac{{\sqrt[3]{3 (t + K)}}^{2}}{3 (t + K)}$

Step 3.3.5.8

Simplify the numerator.

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Step 3.3.5.8.1

Rewrite ${\sqrt[3]{3 (t + K)}}^{2}$ as $\sqrt[3]{{(3 (t + K))}^{2}}$ .

$y = - \frac{\sqrt[3]{{(3 (t + K))}^{2}}}{3 (t + K)}$

Step 3.3.5.8.2

Apply the product rule to $3 (t + K)$ .

$y = - \frac{\sqrt[3]{3^{2} {(t + K)}^{2}}}{3 (t + K)}$

Step 3.3.5.8.3

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

$y = - \frac{\sqrt[3]{9 {(t + K)}^{2}}}{3 (t + K)}$