Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1

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 1.1.1.2

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$1 - x^{- 2}$

Step 1.1.3

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

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

Step 1.1.4

Reorder terms.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Subtract $1$ from both sides of the equation.

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

Step 2.3

Find the LCD of the terms in the equation.

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Step 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 2.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

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

Step 2.4.2.1.2

Cancel the common factor.

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

Step 2.4.2.1.3

Rewrite the expression.

$- 1 = - x^{2}$

Step 2.5

Solve the equation.

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Step 2.5.1

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

$- x^{2} = - 1$

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.5.2.2.2

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

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

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 2.5.3

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

$x = \pm \sqrt{1}$

Step 2.5.4

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.5.5

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

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Step 2.5.5.1

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

$x = 1$

Step 2.5.5.2

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

$x = - 1$

Step 2.5.5.3

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

$x = 1, - 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

$x^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Evaluate $x + \frac{1}{x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 1$ .

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Step 4.1.1

Substitute $1$ for $x$ .

$(1) + \frac{1}{1}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Divide $1$ by $1$ .

$1 + 1$

Step 4.1.2.2

Add $1$ and $1$ .

$2$

Step 4.2

Evaluate at $x = - 1$ .

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Step 4.2.1

Substitute $- 1$ for $x$ .

$(- 1) + \frac{1}{- 1}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Divide $1$ by $- 1$ .

$- 1 - 1$

Step 4.2.2.2

Subtract $1$ from $- 1$ .

$- 2$

Step 4.3

Evaluate at $x = 0$ .

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Step 4.3.1

Substitute $0$ for $x$ .

$(0) + \frac{1}{0}$

Step 4.3.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.4

List all of the points.

$(1, 2), (- 1, - 2)$

Step 5