Calculus Examples

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

Step 1

Rewrite the equation as $\frac{1 + e^{x}}{1 - e^{x}} = y$ .

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

Step 2

Multiply both sides by $1 - e^{x}$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $1 - e^{x}$ .

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

$1 + e^{x} = y (1 - e^{x})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $y (1 - e^{x})$ .

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Step 3.2.1.1

Apply the distributive property.

$1 + e^{x} = y \cdot 1 + y (- e^{x})$

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

Multiply $y$ by $1$ .

$1 + e^{x} = y + y (- e^{x})$

Step 3.2.1.2.2

Rewrite using the commutative property of multiplication.

$1 + e^{x} = y - y e^{x}$

Step 4

Solve for $x$ .

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Step 4.1

Add $y e^{x}$ to both sides of the equation.

$1 + e^{x} + y e^{x} = y$

Step 4.2

Subtract $1$ from both sides of the equation.

$e^{x} + y e^{x} = y - 1$

Step 4.3

Factor $e^{x}$ out of $e^{x} + y e^{x}$ .

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Step 4.3.1

Multiply by $1$ .

$e^{x} \cdot 1 + y e^{x} = y - 1$

Step 4.3.2

Factor $e^{x}$ out of $y e^{x}$ .

$e^{x} \cdot 1 + e^{x} y = y - 1$

Step 4.3.3

Factor $e^{x}$ out of $e^{x} \cdot 1 + e^{x} y$ .

$e^{x} (1 + y) = y - 1$

Step 4.4

Divide each term in $e^{x} (1 + y) = y - 1$ by $1 + y$ and simplify.

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Step 4.4.1

Divide each term in $e^{x} (1 + y) = y - 1$ by $1 + y$ .

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

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $1 + y$ .

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Step 4.4.2.1.1

Cancel the common factor.

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

Step 4.4.2.1.2

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

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

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Combine the numerators over the common denominator.

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

Step 4.5

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (\frac{y - 1}{1 + y})$

Step 4.6

Expand the left side.

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Step 4.6.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (\frac{y - 1}{1 + y})$

Step 4.6.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (\frac{y - 1}{1 + y})$

Step 4.6.3

Multiply $x$ by $1$ .

$x = \ln (\frac{y - 1}{1 + y})$