Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{x}$ and $g (x) = x - 1$ .

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

Step 1.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$\frac{(x - 1) e^{x} - e^{x} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])}{{(x - 1)}^{2}}$

Step 1.1.3.2

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

$\frac{(x - 1) e^{x} - e^{x} (1 + \frac{d}{d x} [- 1])}{{(x - 1)}^{2}}$

Step 1.1.3.3

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

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

Step 1.1.3.4

Simplify the expression.

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Step 1.1.3.4.1

Add $1$ and $0$ .

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

Step 1.1.3.4.2

Multiply $- 1$ by $1$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

Apply the distributive property.

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

Step 1.1.4.2

Simplify the numerator.

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Step 1.1.4.2.1

Rewrite $- 1 e^{x}$ as $- e^{x}$ .

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

Step 1.1.4.2.2

Subtract $e^{x}$ from $- e^{x}$ .

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

Step 1.1.4.3

Reorder terms.

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

Step 1.1.4.4

Factor $e^{x}$ out of $e^{x} x - 2 e^{x}$ .

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Step 1.1.4.4.1

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

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

Step 1.1.4.4.2

Factor $e^{x}$ out of $- 2 e^{x}$ .

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

Step 1.1.4.4.3

Factor $e^{x}$ out of $e^{x} (x) + e^{x} \cdot - 2$ .

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

Step 1.2

The first derivative of $h (x)$ with respect to $x$ is $\frac{e^{x} (x - 2)}{{(x - 1)}^{2}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{e^{x} (x - 2)}{{(x - 1)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

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

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$e^{x} = 0$

$x - 2 = 0$

Step 2.3.2

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $e^{x}$ equal to $0$ .

$e^{x} = 0$

Step 2.3.2.2

Solve $e^{x} = 0$ for $x$ .

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Step 2.3.2.2.1

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

$\ln (e^{x}) = \ln (0)$

Step 2.3.2.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.3.2.2.3

There is no solution for $e^{x} = 0$

No solution

Step 2.3.3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.3.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.3.4

The final solution is all the values that make $e^{x} (x - 2) = 0$ true.

$x = 2$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Evaluate $\frac{e^{x}}{x - 1}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 2$ .

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Step 4.1.1

Substitute $2$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Subtract $1$ from $2$ .

$\frac{e^{2}}{1}$

Step 4.1.2.2

Divide $e^{2}$ by $1$ .

$e^{2}$

Step 4.2

Evaluate at $x = 1$ .

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Step 4.2.1

Substitute $1$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Subtract $1$ from $1$ .

$\frac{e^{1}}{0}$

Step 4.2.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(2, e^{2})$

Step 5