Calculus Examples

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

Step 1

Write $\frac{(\ln (x))}{x}$ as a function.

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

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 2.1.2

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 2.1.3

Differentiate using the Power Rule.

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Step 2.1.3.1

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

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

Step 2.1.3.2

Cancel the common factor of $x$ .

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Step 2.1.3.2.1

Cancel the common factor.

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

Step 2.1.3.2.2

Rewrite the expression.

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

Step 2.1.3.3

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 2.1.3.4

Multiply $- 1$ by $1$ .

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

Step 2.2

Find the second derivative.

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Step 2.2.1

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

Step 2.2.2

Differentiate.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

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

Step 2.2.2.1.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [1 - \ln (x)] - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.2.2

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

$\frac{x^{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- \ln (x)]) - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.2.3

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

$\frac{x^{2} (0 + \frac{d}{d x} [- \ln (x)]) - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.2.4

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

$\frac{x^{2} \frac{d}{d x} [- \ln (x)] - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.2.5

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

$\frac{x^{2} (- \frac{d}{d x} [\ln (x)]) - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.3

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

$\frac{x^{2} (- \frac{1}{x}) - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4

Differentiate using the Power Rule.

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Step 2.2.4.1

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

$\frac{- \frac{x^{2}}{x} - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4.2

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

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Step 2.2.4.2.1

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

$\frac{- \frac{x \cdot x}{x} - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4.2.2

Cancel the common factors.

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Step 2.2.4.2.2.1

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

$\frac{- \frac{x \cdot x}{x^{1}} - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4.2.2.2

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

$\frac{- \frac{x \cdot x}{x \cdot 1} - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4.2.2.3

Cancel the common factor.

$\frac{- \frac{x \cdot x}{x \cdot 1} - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4.2.2.4

Rewrite the expression.

$\frac{- \frac{x}{1} - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4.2.2.5

Divide $x$ by $1$ .

$\frac{- x - (1 - \ln (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.2.4.3

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

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

Step 2.2.4.4

Simplify with factoring out.

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Step 2.2.4.4.1

Multiply $2$ by $- 1$ .

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

Step 2.2.4.4.2

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

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Step 2.2.4.4.2.1

Factor $x$ out of $- x$ .

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

Step 2.2.4.4.2.2

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

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

Step 2.2.4.4.2.3

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

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

Step 2.2.5

Cancel the common factors.

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Step 2.2.5.1

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

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

Step 2.2.5.2

Cancel the common factor.

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

Step 2.2.5.3

Rewrite the expression.

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

Step 2.2.6

Simplify.

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Step 2.2.6.1

Apply the distributive property.

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

Step 2.2.6.2

Simplify the numerator.

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Step 2.2.6.2.1

Simplify each term.

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Step 2.2.6.2.1.1

Multiply $- 2$ by $1$ .

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

Step 2.2.6.2.1.2

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

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Step 2.2.6.2.1.2.1

Multiply $- 1$ by $- 2$ .

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

Step 2.2.6.2.1.2.2

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

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

Step 2.2.6.2.2

Subtract $2$ from $- 1$ .

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

Step 2.2.6.3

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

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

Step 2.2.6.4

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

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

Step 2.2.6.5

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

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

Step 2.2.6.6

Move the negative in front of the fraction.

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

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{3 - \ln (x^{2})}{x^{3}}$ .

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

Step 3

Set the second derivative equal to $0$ then solve the equation $- \frac{3 - \ln (x^{2})}{x^{3}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$- \frac{3 - \ln (x^{2})}{x^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$3 - \ln (x^{2}) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

Subtract $3$ from both sides of the equation.

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

Step 3.3.2

Divide each term in $- \ln (x^{2}) = - 3$ by $- 1$ and simplify.

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Step 3.3.2.1

Divide each term in $- \ln (x^{2}) = - 3$ by $- 1$ .

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.3.2.2.2

Divide $\ln (x^{2})$ by $1$ .

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

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $- 3$ by $- 1$ .

$\ln (x^{2}) = 3$

Step 3.3.3

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

$e^{\ln (x^{2})} = e^{3}$

Step 3.3.4

Rewrite $\ln (x^{2}) = 3$ 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 $b^{y} = x$ .

$e^{3} = x^{2}$

Step 3.3.5

Solve for $x$ .

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Step 3.3.5.1

Rewrite the equation as $x^{2} = e^{3}$ .

$x^{2} = e^{3}$

Step 3.3.5.2

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

$x = \pm \sqrt{e^{3}}$

Step 3.3.5.3

Simplify $\pm \sqrt{e^{3}}$ .

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Step 3.3.5.3.1

Factor out $e^{2}$ .

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

Step 3.3.5.3.2

Pull terms out from under the radical.

$x = \pm e \sqrt{e}$

Step 3.3.5.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.5.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = e \sqrt{e}$

Step 3.3.5.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - e \sqrt{e}$

Step 3.3.5.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = e \sqrt{e}, - e \sqrt{e}$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $e \sqrt{e}$ in $f (x) = \frac{\ln (x)}{x}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $e \sqrt{e}$ in the expression.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $\frac{\ln (e \sqrt{e})}{e \sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

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

Step 4.1.2.2

Combine and simplify the denominator.

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Step 4.1.2.2.1

Multiply $\frac{\ln (e \sqrt{e})}{e \sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

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

Step 4.1.2.2.2

Move $\sqrt{e}$ .

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

Step 4.1.2.2.3

Raise $\sqrt{e}$ to the power of $1$ .

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

Step 4.1.2.2.4

Raise $\sqrt{e}$ to the power of $1$ .

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

Step 4.1.2.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (e \sqrt{e}) = \frac{\ln (e \sqrt{e}) \sqrt{e}}{e {\sqrt{e}}^{1 + 1}}$

Step 4.1.2.2.6

Add $1$ and $1$ .

$f (e \sqrt{e}) = \frac{\ln (e \sqrt{e}) \sqrt{e}}{e {\sqrt{e}}^{2}}$

Step 4.1.2.2.7

Rewrite ${\sqrt{e}}^{2}$ as $e$ .

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Step 4.1.2.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{e}$ as $e^{\frac{1}{2}}$ .

$f (e \sqrt{e}) = \frac{\ln (e \sqrt{e}) \sqrt{e}}{e {(e^{\frac{1}{2}})}^{2}}$

Step 4.1.2.2.7.2

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

$f (e \sqrt{e}) = \frac{\ln (e \sqrt{e}) \sqrt{e}}{e e^{\frac{1}{2} \cdot 2}}$

Step 4.1.2.2.7.3

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

$f (e \sqrt{e}) = \frac{\ln (e \sqrt{e}) \sqrt{e}}{e e^{\frac{2}{2}}}$

Step 4.1.2.2.7.4

Cancel the common factor of $2$ .

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Step 4.1.2.2.7.4.1

Cancel the common factor.

$f (e \sqrt{e}) = \frac{\ln (e \sqrt{e}) \sqrt{e}}{e e^{\frac{2}{2}}}$

Step 4.1.2.2.7.4.2

Rewrite the expression.

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

Step 4.1.2.2.7.5

Simplify.

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

Step 4.1.2.3

Simplify the denominator.

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Step 4.1.2.3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (e \sqrt{e}) = \frac{\ln (e \sqrt{e}) \sqrt{e}}{e^{1 + 1}}$

Step 4.1.2.3.2

Add $1$ and $1$ .

$f (e \sqrt{e}) = \frac{\ln (e \sqrt{e}) \sqrt{e}}{e^{2}}$

Step 4.1.2.4

The final answer is $\frac{\ln (e \sqrt{e}) \sqrt{e}}{e^{2}}$ .

$\frac{\ln (e \sqrt{e}) \sqrt{e}}{e^{2}}$

Step 4.2

The point found by substituting $e \sqrt{e}$ in $f (x) = \frac{\ln (x)}{x}$ is $(e \sqrt{e}, \frac{\ln (e \sqrt{e}) \sqrt{e}}{e^{2}})$ . This point can be an inflection point.

$(e \sqrt{e}, \frac{\ln (e \sqrt{e}) \sqrt{e}}{e^{2}})$

Step 4.3

$- e \sqrt{e}$ is not in the domain of $f (x) = \frac{\ln (x)}{x}$ . There is no inflection point at $- e \sqrt{e}$ .

$- e \sqrt{e}$ is not in the domain

Step 4.4

Determine the points that could be inflection points.

$(e \sqrt{e}, \frac{\ln (e \sqrt{e}) \sqrt{e}}{e^{2}})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, e \sqrt{e}) \cup (e \sqrt{e}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, e \sqrt{e})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $4.38168907$ in the expression.

$f'' (4.38168907) = - \frac{3 - \ln ({(4.38168907)}^{2})}{{(4.38168907)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Raise $4.38168907$ to the power of $2$ .

$f'' (4.38168907) = - \frac{3 - \ln (19.1991991)}{{4.38168907}^{3}}$

Step 6.2.2

Raise $4.38168907$ to the power of $3$ .

$f'' (4.38168907) = - \frac{3 - \ln (19.1991991)}{84.12492089}$

Step 6.2.3

Replace $e$ with an approximation.

$f'' (4.38168907) = - \frac{3 - \log_{2.71828182} (19.1991991)}{84.12492089}$

Step 6.2.4

Log base $2.71828182$ of $19.1991991$ is approximately $2.95486856$ .

$f'' (4.38168907) = - \frac{3 - 1 \cdot 2.95486856}{84.12492089}$

Step 6.2.5

Multiply $- 1$ by $2.95486856$ .

$f'' (4.38168907) = - \frac{3 - 2.95486856}{84.12492089}$

Step 6.2.6

Subtract $2.95486856$ from $3$ .

$f'' (4.38168907) = - \frac{0.04513143}{84.12492089}$

Step 6.2.7

Divide $0.04513143$ by $84.12492089$ .

$f'' (4.38168907) = - 1 \cdot 0.00053648$

Step 6.2.8

Multiply $- 1$ by $0.00053648$ .

$f'' (4.38168907) = - 0.00053648$

Step 6.2.9

The final answer is $- 0.00053648$ .

$- 0.00053648$

Step 6.3

At $4.38168907$ , the second derivative is $- 0.00053648$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, e \sqrt{e})$

Decreasing on $(- \infty, e \sqrt{e})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(e \sqrt{e}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $4.58168907$ in the expression.

$f'' (4.58168907) = - \frac{3 - \ln ({(4.58168907)}^{2})}{{(4.58168907)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Raise $4.58168907$ to the power of $2$ .

$f'' (4.58168907) = - \frac{3 - \ln (20.99187473)}{{4.58168907}^{3}}$

Step 7.2.2

Raise $4.58168907$ to the power of $3$ .

$f'' (4.58168907) = - \frac{3 - \ln (20.99187473)}{96.17824304}$

Step 7.2.3

Replace $e$ with an approximation.

$f'' (4.58168907) = - \frac{3 - \log_{2.71828182} (20.99187473)}{96.17824304}$

Step 7.2.4

Log base $2.71828182$ of $20.99187473$ is approximately $3.04413544$ .

$f'' (4.58168907) = - \frac{3 - 1 \cdot 3.04413544}{96.17824304}$

Step 7.2.5

Multiply $- 1$ by $3.04413544$ .

$f'' (4.58168907) = - \frac{3 - 3.04413544}{96.17824304}$

Step 7.2.6

Subtract $3.04413544$ from $3$ .

$f'' (4.58168907) = - \frac{- 0.04413544}{96.17824304}$

Step 7.2.7

Divide $- 0.04413544$ by $96.17824304$ .

$f'' (4.58168907) = 0.00045889$

Step 7.2.8

The final answer is $0.00045889$ .

$0.00045889$

Step 7.3

At $4.58168907$ , the second derivative is $0.00045889$ . Since this is positive, the second derivative is increasing on the interval $(e \sqrt{e}, \infty)$ .

Increasing on $(e \sqrt{e}, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(e \sqrt{e}, \frac{\ln (e \sqrt{e}) \sqrt{e}}{e^{2}})$ .

$(e \sqrt{e}, \frac{\ln (e \sqrt{e}) \sqrt{e}}{e^{2}})$

Step 9