Calculus Examples

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

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

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $2$ by $- 2$ .

$\frac{1}{x} - 4 x$

Step 1.1.4

Reorder terms.

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 4 x + \frac{1}{x}$ .

$- 4 x + \frac{1}{x}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 4 x + \frac{1}{x} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 4 x + \frac{1}{x} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$1, x, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x$

Step 2.3

Multiply each term in $- 4 x + \frac{1}{x} = 0$ by $x$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $- 4 x + \frac{1}{x} = 0$ by $x$ .

$- 4 x \cdot x + \frac{1}{x} x = 0 x$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.3.2.1.1.1

Move $x$ .

$- 4 (x \cdot x) + \frac{1}{x} x = 0 x$

Step 2.3.2.1.1.2

Multiply $x$ by $x$ .

$- 4 x^{2} + \frac{1}{x} x = 0 x$

Step 2.3.2.1.2

Cancel the common factor of $x$ .

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Step 2.3.2.1.2.1

Cancel the common factor.

$- 4 x^{2} + \frac{1}{x} x = 0 x$

Step 2.3.2.1.2.2

Rewrite the expression.

$- 4 x^{2} + 1 = 0 x$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x$ .

$- 4 x^{2} + 1 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $1$ from both sides of the equation.

$- 4 x^{2} = - 1$

Step 2.4.2

Divide each term in $- 4 x^{2} = - 1$ by $- 4$ and simplify.

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Step 2.4.2.1

Divide each term in $- 4 x^{2} = - 1$ by $- 4$ .

$\frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{- 4 x^{2}}{- 4} = \frac{- 1}{- 4}$

Step 2.4.2.2.1.2

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

$x^{2} = \frac{- 1}{- 4}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{1}{4}$

Step 2.4.3

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

$x = \pm \sqrt{\frac{1}{4}}$

Step 2.4.4

Simplify $\pm \sqrt{\frac{1}{4}}$ .

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Step 2.4.4.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 2.4.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{4}}$

Step 2.4.4.3

Simplify the denominator.

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Step 2.4.4.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.4.4.3.2

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

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

Step 2.4.5

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

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Step 2.4.5.1

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

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

Step 2.4.5.2

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

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

Step 2.4.5.3

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

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4

Evaluate $\ln (x) - 2 x^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{1}{2}$ .

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Step 4.1.1

Substitute $\frac{1}{2}$ for $x$ .

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

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 4.1.2.2

One to any power is one.

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Factor $2$ out of $- 2$ .

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

Step 4.1.2.4.2

Factor $2$ out of $4$ .

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

Step 4.1.2.4.3

Cancel the common factor.

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

Step 4.1.2.4.4

Rewrite the expression.

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

Step 4.1.2.5

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

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

Step 4.2

Evaluate at $x = - \frac{1}{2}$ .

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Step 4.2.1

Substitute $- \frac{1}{2}$ for $x$ .

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

Step 4.2.2

The natural logarithm of a negative number is undefined.

Undefined

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

$\ln (0) - 2 {(0)}^{2}$

Step 4.3.2

The natural logarithm of zero is undefined.

Undefined

Step 4.4

List all of the points.

$(\frac{1}{2}, \ln (\frac{1}{2}) - \frac{1}{2})$

Step 5