Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 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 1.2

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 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $x$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 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 1.3.4

Multiply $- 1$ by $1$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

Step 2.2

Differentiate.

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Step 2.2.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.1.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{x^{2} \frac{d}{d x} (1 - \ln (x)) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.2.2

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

$f'' (x) = \frac{x^{2} (\frac{d}{d x} (1) + \frac{d}{d x} (- \ln (x))) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.2.3

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

$f'' (x) = \frac{x^{2} (0 + \frac{d}{d x} (- \ln (x))) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.2.4

Add $0$ and $\frac{d}{d x} [- \ln (x)]$ .

$f'' (x) = \frac{x^{2} \frac{d}{d x} (- \ln (x)) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (x)$ with respect to $x$ is $- \frac{d}{d x} [\ln (x)]$ .

$f'' (x) = \frac{x^{2} (- \frac{d}{d x} \ln (x)) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.3

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

$f'' (x) = \frac{x^{2} (- \frac{1}{x}) - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4

Differentiate using the Power Rule.

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Step 2.4.1

Combine $x^{2}$ and $\frac{1}{x}$ .

$f'' (x) = \frac{- \frac{x^{2}}{x} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.4.2.1

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

$f'' (x) = \frac{- \frac{x \cdot x}{x} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.2.2

Cancel the common factors.

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Step 2.4.2.2.1

Raise $x$ to the power of $1$ .

$f'' (x) = \frac{- \frac{x \cdot x}{x} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.2.2.2

Factor $x$ out of $x^{1}$ .

$f'' (x) = \frac{- \frac{x \cdot x}{x \cdot 1} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.2.2.3

Cancel the common factor.

$f'' (x) = \frac{- \frac{x \cdot x}{x \cdot 1} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.2.2.4

Rewrite the expression.

$f'' (x) = \frac{- \frac{x}{1} - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.2.2.5

Divide $x$ by $1$ .

$f'' (x) = \frac{- x - (1 - \ln (x)) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = \frac{- x - (1 - \ln (x)) (2 x)}{x^{4}}$

Step 2.4.4

Simplify with factoring out.

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Step 2.4.4.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{- x - 2 (1 - \ln (x)) x}{x^{4}}$

Step 2.4.4.2

Factor $x$ out of $- x - 2 (1 - \ln (x)) x$ .

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Step 2.4.4.2.1

Factor $x$ out of $- x$ .

$f'' (x) = \frac{x \cdot - 1 - 2 (1 - \ln (x)) x}{x^{4}}$

Step 2.4.4.2.2

Factor $x$ out of $- 2 (1 - \ln (x)) x$ .

$f'' (x) = \frac{x \cdot - 1 + x (- 2 (1 - \ln (x)))}{x^{4}}$

Step 2.4.4.2.3

Factor $x$ out of $x \cdot - 1 + x (- 2 (1 - \ln (x)))$ .

$f'' (x) = \frac{x (- 1 - 2 (1 - \ln (x)))}{x^{4}}$

Step 2.5

Cancel the common factors.

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Step 2.5.1

Factor $x$ out of $x^{4}$ .

$f'' (x) = \frac{x (- 1 - 2 (1 - \ln (x)))}{x \cdot x^{3}}$

Step 2.5.2

Cancel the common factor.

$f'' (x) = \frac{x (- 1 - 2 (1 - \ln (x)))}{x \cdot x^{3}}$

Step 2.5.3

Rewrite the expression.

$f'' (x) = \frac{- 1 - 2 (1 - \ln (x))}{x^{3}}$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$f'' (x) = \frac{- 1 - 2 \cdot 1 - 2 (- \ln (x))}{x^{3}}$

Step 2.6.2

Simplify the numerator.

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Step 2.6.2.1

Simplify each term.

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Step 2.6.2.1.1

Multiply $- 2$ by $1$ .

$f'' (x) = \frac{- 1 - 2 - 2 (- \ln (x))}{x^{3}}$

Step 2.6.2.1.2

Multiply $- 2 (- \ln (x))$ .

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Step 2.6.2.1.2.1

Multiply $- 1$ by $- 2$ .

$f'' (x) = \frac{- 1 - 2 + 2 \ln (x)}{x^{3}}$

Step 2.6.2.1.2.2

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

$f'' (x) = \frac{- 1 - 2 + \ln (x^{2})}{x^{3}}$

Step 2.6.2.2

Subtract $2$ from $- 1$ .

$f'' (x) = \frac{- 3 + \ln (x^{2})}{x^{3}}$

Step 2.6.3

Rewrite $- 3$ as $- 1 (3)$ .

$f'' (x) = \frac{- 1 \cdot 3 + \ln (x^{2})}{x^{3}}$

Step 2.6.4

Factor $- 1$ out of $\ln (x^{2})$ .

$f'' (x) = \frac{- 1 \cdot 3 - 1 (- \ln (x^{2}))}{x^{3}}$

Step 2.6.5

Factor $- 1$ out of $- 1 (3) - 1 (- \ln (x^{2}))$ .

$f'' (x) = \frac{- 1 (3 - \ln (x^{2}))}{x^{3}}$

Step 2.6.6

Move the negative in front of the fraction.

$f'' (x) = - \frac{3 - \ln (x^{2})}{x^{3}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

Step 4.1.2

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 4.1.3

Differentiate using the Power Rule.

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Step 4.1.3.1

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

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

Step 4.1.3.2

Cancel the common factor of $x$ .

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Step 4.1.3.2.1

Cancel the common factor.

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

Step 4.1.3.2.2

Rewrite the expression.

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

Step 4.1.3.3

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 4.1.3.4

Multiply $- 1$ by $1$ .

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$1 - \ln (x) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Subtract $1$ from both sides of the equation.

$- \ln (x) = - 1$

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.3.2.2.2

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

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

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $- 1$ by $- 1$ .

$\ln (x) = 1$

Step 5.3.3

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

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

Step 5.3.4

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 5.3.5

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

$x = e$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$x^{2} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

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

$x = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

$x = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.3

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

$x \leq 0$

Step 6.4

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 7

Critical points to evaluate.

$x = e$

Step 8

Evaluate the second derivative at $x = e$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{3 - \ln ({(e)}^{2})}{{(e)}^{3}}$

Step 9

Simplify the numerator.

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Step 9.1

Use logarithm rules to move $2$ out of the exponent.

$- \frac{3 - (2 \ln (e))}{e^{3}}$

Step 9.2

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

$- \frac{3 - (2 \cdot 1)}{e^{3}}$

Step 9.3

Multiply $2$ by $1$ .

$- \frac{3 - 1 \cdot 2}{e^{3}}$

Step 9.4

Multiply $- 1$ by $2$ .

$- \frac{3 - 2}{e^{3}}$

Step 9.5

Subtract $2$ from $3$ .

$- \frac{1}{e^{3}}$

Step 10

$x = e$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = e$ is a local maximum

Step 11

Find the y-value when $x = e$ .

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Step 11.1

Replace the variable $x$ with $e$ in the expression.

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

Step 11.2

Simplify the result.

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Step 11.2.1

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

$f (e) = \frac{1}{e}$

Step 11.2.2

The final answer is $\frac{1}{e}$ .

$y = \frac{1}{e}$

Step 12

These are the local extrema for $f (x) = \frac{\ln (x)}{x}$ .

$(e, \frac{1}{e})$ is a local maxima

Step 13