Calculus Examples

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

Step 1

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

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

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Differentiate.

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Step 2.1.1.1.1

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

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

Step 2.1.1.1.2

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

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

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

Multiply $- 1$ by $1$ .

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

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

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

Step 2.1.1.4

Reorder terms.

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

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

By the Sum Rule, the derivative of $2 x - \frac{1}{x} - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [- 1]$ .

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

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

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

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

Step 2.1.2.2.3

Multiply $2$ by $1$ .

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

Step 2.1.2.3

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

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Step 2.1.2.3.1

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 $g (x) = \frac{1}{x}$ .

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

Step 2.1.2.3.2

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

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

Step 2.1.2.3.3

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

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

Step 2.1.2.3.4

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

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

Step 2.1.2.3.5

Multiply $- 1$ by $- 1$ .

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

Step 2.1.2.3.6

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

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

Step 2.1.2.3.7

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

$2 + x^{- 2} + 0 + \frac{d}{d x} [- 1]$

Step 2.1.2.3.8

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

Simplify.

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Step 2.1.2.5.1

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

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

Step 2.1.2.5.2

Add $2 + \frac{1}{x^{2}}$ and $0$ .

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

Step 2.1.2.5.3

Reorder terms.

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

Step 2.1.3

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

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

Step 2.2

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

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Step 2.2.1

Set the second derivative equal to $0$ .

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

Step 2.2.2

Subtract $2$ from both sides of the equation.

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

Step 2.2.3

Find the LCD of the terms in the equation.

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Step 2.2.3.1

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 2.2.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 2.2.4

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

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Step 2.2.4.1

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

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

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

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

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Step 2.2.4.2.1.1

Cancel the common factor.

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

Step 2.2.4.2.1.2

Rewrite the expression.

$1 = - 2 x^{2}$

Step 2.2.5

Solve the equation.

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Step 2.2.5.1

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

$- 2 x^{2} = 1$

Step 2.2.5.2

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

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Step 2.2.5.2.1

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

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

Step 2.2.5.2.2

Simplify the left side.

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Step 2.2.5.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.2.5.2.2.1.1

Cancel the common factor.

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

Step 2.2.5.2.2.1.2

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

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

Step 2.2.5.2.3

Simplify the right side.

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Step 2.2.5.2.3.1

Move the negative in front of the fraction.

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

Step 2.2.5.3

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

$x = \pm \sqrt{- \frac{1}{2}}$

Step 2.2.5.4

Simplify $\pm \sqrt{- \frac{1}{2}}$ .

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Step 2.2.5.4.1

Rewrite $- \frac{1}{2}$ as $i^{2} \frac{1^{2}}{2}$ .

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Step 2.2.5.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1}{2}}$

Step 2.2.5.4.1.2

Rewrite $1$ as $1^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1^{2}}{2}}$

Step 2.2.5.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{2}}$

Step 2.2.5.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{2}}$

Step 2.2.5.4.4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{2}}$

Step 2.2.5.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{2}}$

Step 2.2.5.4.6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 2.2.5.4.7

Combine and simplify the denominator.

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Step 2.2.5.4.7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.2.5.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.2.5.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.2.5.4.7.4

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

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.2.5.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.2.5.4.7.6

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

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Step 2.2.5.4.7.6.1

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

$x = \pm i \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.2.5.4.7.6.2

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

$x = \pm i \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.2.5.4.7.6.3

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

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.2.5.4.7.6.4

Cancel the common factor of $2$ .

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Step 2.2.5.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.2.5.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{2}}{2^{1}}$

Step 2.2.5.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2}}{2}$

Step 2.2.5.4.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$x = \pm \frac{i \sqrt{2}}{2}$

Step 2.2.5.5

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

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Step 2.2.5.5.1

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

$x = \frac{i \sqrt{2}}{2}$

Step 2.2.5.5.2

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

$x = - \frac{i \sqrt{2}}{2}$

Step 2.2.5.5.3

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

$x = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}$

Step 3

Find the domain of $f (x) = x^{2} - x - \ln (x)$ .

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Step 3.1

Set the argument in $\ln (x)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 3.2

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(0, \infty)$

Step 5

Substitute any number from the interval $(0, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $2$ in the expression.

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

Step 5.2

Simplify the result.

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Step 5.2.1

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

$f'' (2) = \frac{1}{4} + 2$

Step 5.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f'' (2) = \frac{1}{4} + 2 \cdot \frac{4}{4}$

Step 5.2.3

Combine $2$ and $\frac{4}{4}$ .

$f'' (2) = \frac{1}{4} + \frac{2 \cdot 4}{4}$

Step 5.2.4

Combine the numerators over the common denominator.

$f'' (2) = \frac{1 + 2 \cdot 4}{4}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $2$ by $4$ .

$f'' (2) = \frac{1 + 8}{4}$

Step 5.2.5.2

Add $1$ and $8$ .

$f'' (2) = \frac{9}{4}$

Step 5.2.6

The final answer is $\frac{9}{4}$ .

$\frac{9}{4}$

Step 5.3

The graph is concave up on the interval $(0, \infty)$ because $f'' (2)$ is positive.

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 6