Calculus Examples

$\frac{d y}{d x} = {(\frac{1 + x}{1 + y})}^{2}$

Step 1

Separate the variables.

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Step 1.1

Apply the product rule to $\frac{1 + x}{1 + y}$ .

$\frac{d y}{d x} = \frac{{(1 + x)}^{2}}{{(1 + y)}^{2}}$

Step 1.2

Multiply both sides by ${(1 + y)}^{2}$ .

${(1 + y)}^{2} \frac{d y}{d x} = {(1 + y)}^{2} \frac{{(1 + x)}^{2}}{{(1 + y)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of ${(1 + y)}^{2}$ .

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Step 1.3.1.1

Cancel the common factor.

${(1 + y)}^{2} \frac{d y}{d x} = {(1 + y)}^{2} \frac{{(1 + x)}^{2}}{{(1 + y)}^{2}}$

Step 1.3.1.2

Rewrite the expression.

${(1 + y)}^{2} \frac{d y}{d x} = {(1 + x)}^{2}$

Step 1.3.2

Rewrite ${(1 + x)}^{2}$ as $(1 + x) (1 + x)$ .

${(1 + y)}^{2} \frac{d y}{d x} = (1 + x) (1 + x)$

Step 1.3.3

Expand $(1 + x) (1 + x)$ using the FOIL Method.

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Step 1.3.3.1

Apply the distributive property.

${(1 + y)}^{2} \frac{d y}{d x} = 1 (1 + x) + x (1 + x)$

Step 1.3.3.2

Apply the distributive property.

${(1 + y)}^{2} \frac{d y}{d x} = 1 \cdot 1 + 1 x + x (1 + x)$

Step 1.3.3.3

Apply the distributive property.

${(1 + y)}^{2} \frac{d y}{d x} = 1 \cdot 1 + 1 x + x \cdot 1 + x \cdot x$

Step 1.3.4

Simplify and combine like terms.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $1$ by $1$ .

${(1 + y)}^{2} \frac{d y}{d x} = 1 + 1 x + x \cdot 1 + x \cdot x$

Step 1.3.4.1.2

Multiply $x$ by $1$ .

${(1 + y)}^{2} \frac{d y}{d x} = 1 + x + x \cdot 1 + x \cdot x$

Step 1.3.4.1.3

Multiply $x$ by $1$ .

${(1 + y)}^{2} \frac{d y}{d x} = 1 + x + x + x \cdot x$

Step 1.3.4.1.4

Multiply $x$ by $x$ .

${(1 + y)}^{2} \frac{d y}{d x} = 1 + x + x + x^{2}$

Step 1.3.4.2

Add $x$ and $x$ .

${(1 + y)}^{2} \frac{d y}{d x} = 1 + 2 x + x^{2}$

Step 1.4

Rewrite the equation.

${(1 + y)}^{2} d y = (1 + 2 x + x^{2}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int {(1 + y)}^{2} d y = \int 1 + 2 x + x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Differentiate $1 + y$ .

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

Step 2.2.1.1.2

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

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

Step 2.2.1.1.3

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

$0 + \frac{d}{d y} [y]$

Step 2.2.1.1.4

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

$0 + 1$

Step 2.2.1.1.5

Add $0$ and $1$ .

$1$

Step 2.2.1.2

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

$\int u^{2} d u = \int 1 + 2 x + x^{2} d x$

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 2.3.6.2

Simplify.

$\frac{1}{3} {(1 + y)}^{3} + C_{1} = x + x^{2} + \frac{1}{3} x^{3} + C_{5}$

Step 2.3.7

Reorder terms.

$\frac{1}{3} {(1 + y)}^{3} + C_{1} = x^{2} + \frac{1}{3} x^{3} + x + C_{5}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (x^{2} + \frac{1}{3} x^{3} + x + K)$ .

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.1

Cancel the common factor.

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

Step 3.2.2.1.3.2

Rewrite the expression.

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

Step 3.3

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

$1 + y = \sqrt[3]{3 x^{2} + x^{3} + 3 x + 3 K}$

Step 3.4

Subtract $1$ from both sides of the equation.

$y = \sqrt[3]{3 x^{2} + x^{3} + 3 x + 3 K} - 1$

Step 4

Simplify the constant of integration.

$y = \sqrt[3]{3 x^{2} + x^{3} + 3 x + K} - 1$