Calculus Examples

$f (x) = \frac{\ln (x)}{x}$

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 $f (x) = \ln (x)$ and $g (x) = x$ .

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

Step 1.1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

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

Step 1.1.3

Differentiate using the Power Rule.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Cancel the common factor of $x$ .

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Step 1.1.3.2.1

Cancel the common factor.

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

Step 1.1.3.2.2

Rewrite the expression.

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

Step 1.1.3.3

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

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

Step 1.1.3.4

Multiply $- 1$ by $1$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$1 - \ln (x) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Subtract $1$ from both sides of the equation.

$- \ln (x) = - 1$

Step 2.3.2

Divide each term in $- \ln (x) = - 1$ by $- 1$ and simplify.

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Step 2.3.2.1

Divide each term in $- \ln (x) = - 1$ by $- 1$ .

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

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2.2

Divide $\ln (x)$ by $1$ .

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

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $- 1$ by $- 1$ .

$\ln (x) = 1$

Step 2.3.3

To solve for $x$ , rewrite the equation using properties of logarithms.

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

Step 2.3.4

Rewrite $\ln (x) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{1} = x$

Step 2.3.5

Rewrite the equation as $x = e^{1}$ .

$x = e$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

$x^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.2.1

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

$x = \pm \sqrt{0^{2}}$

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.3

Set the argument in $\ln (x)$ less than or equal to $0$ to find where the expression is undefined.

$x \leq 0$

Step 3.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 4

Evaluate $\frac{\ln (x)}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = e$ .

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Step 4.1.1

Substitute $e$ for $x$ .

$\frac{\ln (e)}{e}$

Step 4.1.2

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

$\frac{1}{e}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$\frac{\ln (0)}{0}$

Step 4.2.2

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

Undefined

Step 4.3

List all of the points.

$(e, \frac{1}{e})$

Step 5