Calculus Examples

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

Step 1

Write $\frac{x^{2}}{2} - \ln (x)$ as a function.

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

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

By the Sum Rule, the derivative of $\frac{x^{2}}{2} - \ln (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [- \ln (x)]$ .

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

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{x^{2}}{2}]$ .

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Step 2.1.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{2}]$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

Cancel the common factor of $2$ .

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Step 2.1.2.5.1

Cancel the common factor.

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

Step 2.1.2.5.2

Divide $x$ by $1$ .

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

Step 2.1.3

Evaluate $\frac{d}{d x} [- \ln (x)]$ .

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.2

Find the second derivative.

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Step 2.2.1

Differentiate.

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Step 2.2.1.1

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

$\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{x}]$

Step 2.2.1.2

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

$1 + \frac{d}{d x} [- \frac{1}{x}]$

Step 2.2.2

Evaluate $\frac{d}{d x} [- \frac{1}{x}]$ .

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Step 2.2.2.1

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

$1 - \frac{d}{d x} [\frac{1}{x}] + \frac{1}{x} \frac{d}{d x} [- 1]$

Step 2.2.2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$1 - \frac{d}{d x} [x^{- 1}] + \frac{1}{x} \frac{d}{d x} [- 1]$

Step 2.2.2.3

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

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

Step 2.2.2.4

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

$1 - - x^{- 2} + \frac{1}{x} \cdot 0$

Step 2.2.2.5

Multiply $- 1$ by $- 1$ .

$1 + 1 x^{- 2} + \frac{1}{x} \cdot 0$

Step 2.2.2.6

Multiply $x^{- 2}$ by $1$ .

$1 + x^{- 2} + \frac{1}{x} \cdot 0$

Step 2.2.2.7

Multiply $\frac{1}{x}$ by $0$ .

$1 + x^{- 2} + 0$

Step 2.2.2.8

Add $x^{- 2}$ and $0$ .

$1 + x^{- 2}$

Step 2.2.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 + \frac{1}{x^{2}}$

Step 2.2.4

Reorder terms.

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

Step 2.3

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

$\frac{1}{x^{2}} + 1$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{1}{x^{2}} + 1 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{1}{x^{2}} + 1 = 0$

Step 3.2

Subtract $1$ from both sides of the equation.

$\frac{1}{x^{2}} = - 1$

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2}, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 3.4

Multiply each term in $\frac{1}{x^{2}} = - 1$ by $x^{2}$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{1}{x^{2}} = - 1$ by $x^{2}$ .

$\frac{1}{x^{2}} x^{2} = - x^{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{1}{x^{2}} x^{2} = - x^{2}$

Step 3.4.2.1.2

Rewrite the expression.

$1 = - x^{2}$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $- x^{2} = 1$ .

$- x^{2} = 1$

Step 3.5.2

Divide each term in $- x^{2} = 1$ by $- 1$ and simplify.

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Step 3.5.2.1

Divide each term in $- x^{2} = 1$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{1}{- 1}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{1}{- 1}$

Step 3.5.2.2.2

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

$x^{2} = \frac{1}{- 1}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Divide $1$ by $- 1$ .

$x^{2} = - 1$

Step 3.5.3

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

$x = \pm \sqrt{- 1}$

Step 3.5.4

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 3.5.5

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

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Step 3.5.5.1

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

$x = i$

Step 3.5.5.2

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

$x = - i$

Step 3.5.5.3

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

$x = i, - i$

Step 4

No values found that can make the second derivative equal to $0$ .

No Inflection Points