Calculus Examples

$\frac{d y}{d x} = - \frac{1}{4} e^{x} y^{- 2}$

Step 1

Separate the variables.

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Step 1.1

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

$\frac{1}{y^{- 2}} \frac{d y}{d x} = \frac{1}{y^{- 2}} (- \frac{1}{4} e^{x} y^{- 2})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{- 2}} \frac{d y}{d x} = - \frac{1}{4} \cdot \frac{1}{y^{- 2}} (e^{x} y^{- 2})$

Step 1.2.2

Move $y^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{1}{y^{- 2}} \frac{d y}{d x} = - \frac{1}{4} y^{2} (e^{x} y^{- 2})$

Step 1.2.3

Multiply $y^{2}$ by $y^{- 2}$ by adding the exponents.

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Step 1.2.3.1

Move $y^{- 2}$ .

$\frac{1}{y^{- 2}} \frac{d y}{d x} = - \frac{1}{4} (y^{- 2} y^{2}) e^{x}$

Step 1.2.3.2

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

$\frac{1}{y^{- 2}} \frac{d y}{d x} = - \frac{1}{4} y^{- 2 + 2} e^{x}$

Step 1.2.3.3

Add $- 2$ and $2$ .

$\frac{1}{y^{- 2}} \frac{d y}{d x} = - \frac{1}{4} y^{0} e^{x}$

Step 1.2.4

Simplify $- \frac{1}{4} y^{0} e^{x}$ .

$\frac{1}{y^{- 2}} \frac{d y}{d x} = - \frac{1}{4} e^{x}$

Step 1.2.5

Combine $e^{x}$ and $\frac{1}{4}$ .

$\frac{1}{y^{- 2}} \frac{d y}{d x} = - \frac{e^{x}}{4}$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{- 2}} d y = - \frac{e^{x}}{4} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{- 2}} d y = \int - \frac{e^{x}}{4} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Move $y^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\int y^{2} d y = \int - \frac{e^{x}}{4} d x$

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.3.2

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

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

Step 2.3.3

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

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

Step 2.3.4

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (- \frac{1}{4} e^{x} + K)$ .

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Step 3.2.2.1.1

Combine $e^{x}$ and $\frac{1}{4}$ .

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Multiply $3 (- \frac{e^{x}}{4})$ .

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Step 3.2.2.1.3.1

Multiply $- 1$ by $3$ .

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

Step 3.2.2.1.3.2

Combine $- 3$ and $\frac{e^{x}}{4}$ .

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

Step 3.2.2.1.4

Move the negative in front of the fraction.

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

Step 3.3

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

$y = \sqrt[3]{- \frac{3 e^{x}}{4} + 3 K}$

Step 3.4

Simplify $\sqrt[3]{- \frac{3 e^{x}}{4} + 3 K}$ .

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Step 3.4.1

Factor $3$ out of $- \frac{3 e^{x}}{4} + 3 K$ .

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Step 3.4.1.1

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

$y = \sqrt[3]{3 (- \frac{e^{x}}{4}) + 3 K}$

Step 3.4.1.2

Factor $3$ out of $3 K$ .

$y = \sqrt[3]{3 (- \frac{e^{x}}{4}) + 3 K}$

Step 3.4.1.3

Factor $3$ out of $3 (- \frac{e^{x}}{4}) + 3 K$ .

$y = \sqrt[3]{3 (- \frac{e^{x}}{4} + K)}$

Step 3.4.2

To write $K$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \sqrt[3]{3 (- \frac{e^{x}}{4} + K \cdot \frac{4}{4})}$

Step 3.4.3

Simplify terms.

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Step 3.4.3.1

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

$y = \sqrt[3]{3 (- \frac{e^{x}}{4} + \frac{K \cdot 4}{4})}$

Step 3.4.3.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{3 \frac{- e^{x} + K \cdot 4}{4}}$

Step 3.4.4

Move $4$ to the left of $K$ .

$y = \sqrt[3]{3 \frac{- e^{x} + 4 K}{4}}$

Step 3.4.5

Combine $3$ and $\frac{- e^{x} + 4 K}{4}$ .

$y = \sqrt[3]{\frac{3 (- e^{x} + 4 K)}{4}}$

Step 3.4.6

Rewrite $\sqrt[3]{\frac{3 (- e^{x} + 4 K)}{4}}$ as $\frac{\sqrt[3]{3 (- e^{x} + 4 K)}}{\sqrt[3]{4}}$ .

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)}}{\sqrt[3]{4}}$

Step 3.4.7

Multiply $\frac{\sqrt[3]{3 (- e^{x} + 4 K)}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 3.4.8

Combine and simplify the denominator.

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Step 3.4.8.1

Multiply $\frac{\sqrt[3]{3 (- e^{x} + 4 K)}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

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

Step 3.4.8.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 3.4.8.3

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

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 3.4.8.4

Add $1$ and $2$ .

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

Step 3.4.8.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 3.4.8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 3.4.8.5.2

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

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 3.4.8.5.3

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

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

Step 3.4.8.5.4

Cancel the common factor of $3$ .

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Step 3.4.8.5.4.1

Cancel the common factor.

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

Step 3.4.8.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 3.4.8.5.5

Evaluate the exponent.

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

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

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

Step 3.4.9.2

Raise $4$ to the power of $2$ .

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)} \sqrt[3]{16}}{4}$

Step 3.4.9.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 3.4.9.3.1

Factor $8$ out of $16$ .

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)} \sqrt[3]{8 (2)}}{4}$

Step 3.4.9.3.2

Rewrite $8$ as $2^{3}$ .

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)} \sqrt[3]{2^{3} \cdot 2}}{4}$

Step 3.4.9.4

Pull terms out from under the radical.

$y = \frac{\sqrt[3]{3 (- e^{x} + 4 K)} \cdot 2 \sqrt[3]{2}}{4}$

Step 3.4.9.5

Combine exponents.

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Step 3.4.9.5.1

Combine using the product rule for radicals.

$y = \frac{2 \sqrt[3]{3 (- e^{x} + 4 K) \cdot 2}}{4}$

Step 3.4.9.5.2

Multiply $2$ by $3$ .

$y = \frac{2 \sqrt[3]{6 (- e^{x} + 4 K)}}{4}$

Step 3.4.10

Cancel the common factors.

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Step 3.4.10.1

Factor $2$ out of $4$ .

$y = \frac{2 \sqrt[3]{6 (- e^{x} + 4 K)}}{2 \cdot 2}$

Step 3.4.10.2

Cancel the common factor.

$y = \frac{2 \sqrt[3]{6 (- e^{x} + 4 K)}}{2 \cdot 2}$

Step 3.4.10.3

Rewrite the expression.

$y = \frac{\sqrt[3]{6 (- e^{x} + 4 K)}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{6 (- e^{x} + K)}}{2}$