Calculus Examples

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

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 $2 x - \frac{1}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{x^{2}}]$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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 $n = 1$ .

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

Step 1.1.2.3

Multiply $2$ by $1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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 $g (x) = \frac{1}{x^{2}}$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 1.1.3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

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

Step 1.1.3.3.2

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

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

Step 1.1.3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 1.1.3.4

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

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

Step 1.1.3.5

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

$f' (x) = 2 - (- {(x^{2})}^{- 2} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 1.1.3.6

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

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Step 1.1.3.6.1

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

$f' (x) = 2 - (- x^{2 \cdot - 2} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 1.1.3.6.2

Multiply $2$ by $- 2$ .

$f' (x) = 2 - (- x^{- 4} (2 x)) + \frac{1}{x^{2}} \cdot 0$

Step 1.1.3.7

Multiply $2$ by $- 1$ .

$f' (x) = 2 - (- 2 x^{- 4} x) + \frac{1}{x^{2}} \cdot 0$

Step 1.1.3.8

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

$f' (x) = 2 - (- 2 (x \cdot x^{- 4})) + \frac{1}{x^{2}} \cdot 0$

Step 1.1.3.9

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

$f' (x) = 2 - (- 2 x^{1 - 4}) + \frac{1}{x^{2}} \cdot 0$

Step 1.1.3.10

Subtract $4$ from $1$ .

$f' (x) = 2 - (- 2 x^{- 3}) + \frac{1}{x^{2}} \cdot 0$

Step 1.1.3.11

Multiply $- 2$ by $- 1$ .

$f' (x) = 2 + 2 x^{- 3} + \frac{1}{x^{2}} \cdot 0$

Step 1.1.3.12

Multiply $\frac{1}{x^{2}}$ by $0$ .

$f' (x) = 2 + 2 x^{- 3} + 0$

Step 1.1.3.13

Add $2 x^{- 3}$ and $0$ .

$f' (x) = 2 + 2 x^{- 3}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

$f' (x) = 2 + 2 (\frac{1}{x^{3}})$

Step 1.1.4.2

Combine $2$ and $\frac{1}{x^{3}}$ .

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

Step 1.1.4.3

Reorder terms.

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

Step 1.2

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

$\frac{2}{x^{3}} + 2$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{2}{x^{3}} + 2 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{2}{x^{3}} + 2 = 0$

Step 2.2

Subtract $2$ from both sides of the equation.

$\frac{2}{x^{3}} = - 2$

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.

$x^{3}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 2.4

Multiply each term in $\frac{2}{x^{3}} = - 2$ by $x^{3}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{2}{x^{3}} = - 2$ by $x^{3}$ .

$\frac{2}{x^{3}} x^{3} = - 2 x^{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $x^{3}$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{2}{x^{3}} x^{3} = - 2 x^{3}$

Step 2.4.2.1.2

Rewrite the expression.

$2 = - 2 x^{3}$

Step 2.5

Solve the equation.

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Step 2.5.1

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

$- 2 x^{3} = 2$

Step 2.5.2

Subtract $2$ from both sides of the equation.

$- 2 x^{3} - 2 = 0$

Step 2.5.3

Factor the left side of the equation.

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Step 2.5.3.1

Factor $- 2$ out of $- 2 x^{3} - 2$ .

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Step 2.5.3.1.1

Factor $- 2$ out of $- 2 x^{3}$ .

$- 2 x^{3} - 2 = 0$

Step 2.5.3.1.2

Factor $- 2$ out of $- 2$ .

$- 2 x^{3} - 2 \cdot 1 = 0$

Step 2.5.3.1.3

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

$- 2 (x^{3} + 1) = 0$

Step 2.5.3.2

Rewrite $1$ as $1^{3}$ .

$- 2 (x^{3} + 1^{3}) = 0$

Step 2.5.3.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

$- 2 ((x + 1) (x^{2} - x \cdot 1 + 1^{2})) = 0$

Step 2.5.3.4

Factor.

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Step 2.5.3.4.1

Simplify.

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Step 2.5.3.4.1.1

Multiply $- 1$ by $1$ .

$- 2 ((x + 1) (x^{2} - x + 1^{2})) = 0$

Step 2.5.3.4.1.2

One to any power is one.

$- 2 ((x + 1) (x^{2} - x + 1)) = 0$

Step 2.5.3.4.2

Remove unnecessary parentheses.

$- 2 (x + 1) (x^{2} - x + 1) = 0$

Step 2.5.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

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

Step 2.5.5

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.5.5.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.5.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5.6

Set $x^{2} - x + 1$ equal to $0$ and solve for $x$ .

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Step 2.5.6.1

Set $x^{2} - x + 1$ equal to $0$ .

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

Step 2.5.6.2

Solve $x^{2} - x + 1 = 0$ for $x$ .

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Step 2.5.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5.6.2.2

Substitute the values $a = 1$ , $b = - 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 2.5.6.2.3

Simplify.

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Step 2.5.6.2.3.1

Simplify the numerator.

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Step 2.5.6.2.3.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.5.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.5.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.5.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.5.6.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.5.6.2.3.1.4

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

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.6.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.5.6.2.3.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.5.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 2.5.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.5.6.2.4.1

Simplify the numerator.

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Step 2.5.6.2.4.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.5.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.5.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.5.6.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.5.6.2.4.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.5.6.2.4.1.4

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

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.6.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.5.6.2.4.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.5.6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 2.5.6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{3}}{2}$

Step 2.5.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.6.2.5.1

Simplify the numerator.

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Step 2.5.6.2.5.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.5.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.5.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.5.6.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.5.6.2.5.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.5.6.2.5.1.4

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

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.6.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.5.6.2.5.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.5.6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 2.5.6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{3}}{2}$

Step 2.5.6.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 2.5.7

The final solution is all the values that make $- 2 (x + 1) (x^{2} - x + 1) = 0$ true.

$x = - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{2}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

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.

$x = \sqrt[3]{0}$

Step 3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 3.2.2.2

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

$x = 0$

Step 4

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

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Multiply $2$ by $- 1$ .

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

Step 4.1.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 4.1.2.1.3

Cancel the common factor of $1$ .

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Step 4.1.2.1.3.1

Cancel the common factor.

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

Step 4.1.2.1.3.2

Rewrite the expression.

$- 2 - 1 \cdot 1$

Step 4.1.2.1.4

Multiply $- 1$ by $1$ .

$- 2 - 1$

Step 4.1.2.2

Subtract $1$ from $- 2$ .

$- 3$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$2 (0) - \frac{1}{{(0)}^{2}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raising $0$ to any positive power yields $0$ .

$2 (0) - \frac{1}{0}$

Step 4.2.2.2

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

Undefined

Step 4.3

List all of the points.

$(- 1, - 3)$

Step 5