Calculus Examples

$\frac{e^{x}}{{(1 - e^{x})}^{2}}$

Step 1

Set the denominator in $\frac{e^{x}}{{(1 - e^{x})}^{2}}$ equal to $0$ to find where the expression is undefined.

${(1 - e^{x})}^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$1 - e^{x} = \pm \sqrt{0}$

Step 2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.1

Rewrite $0$ as $0^{2}$ .

$1 - e^{x} = \pm \sqrt{0^{2}}$

Step 2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$1 - e^{x} = \pm 0$

Step 2.2.3

Plus or minus $0$ is $0$ .

$1 - e^{x} = 0$

Step 2.3

Subtract $1$ from both sides of the equation.

$- e^{x} = - 1$

Step 2.4

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

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Step 2.4.1

Divide each term in $- e^{x} = - 1$ by $- 1$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

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

Step 2.4.2.2

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

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $- 1$ by $- 1$ .

$e^{x} = 1$

Step 2.5

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

$\ln (e^{x}) = \ln (1)$

Step 2.6

Expand the left side.

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Step 2.6.1

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

$x \ln (e) = \ln (1)$

Step 2.6.2

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

$x \cdot 1 = \ln (1)$

Step 2.6.3

Multiply $x$ by $1$ .

$x = \ln (1)$

Step 2.7

The natural logarithm of $1$ is $0$ .

$x = 0$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4