Finite Math Examples

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

Step 1

Set $\frac{- 1}{{(x - 1)}^{2}} - \frac{1}{{(x - 1)}^{2}} \cdot e^{\frac{1}{x - 1}}$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Move the negative in front of the fraction.

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

Step 2.1.1.2

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

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

Step 2.1.2

Simplify terms.

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Step 2.1.2.1

Combine the numerators over the common denominator.

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

Step 2.1.2.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.1.2.3

Factor $- 1$ out of $- e^{\frac{1}{x - 1}}$ .

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

Step 2.1.2.4

Factor $- 1$ out of $- 1 (1) - (e^{\frac{1}{x - 1}})$ .

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

Step 2.1.2.5

Move the negative in front of the fraction.

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

Step 2.2

Set the numerator equal to zero.

$1 + e^{\frac{1}{x - 1}} = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Subtract $1$ from both sides of the equation.

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

Step 2.3.2

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

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

Step 2.3.3

The equation cannot be solved because $\ln (- 1)$ is undefined.

Undefined

Step 2.3.4

There is no solution for $e^{\frac{1}{x - 1}} = - 1$

No solution