Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1

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

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

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 $n = 2$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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 1.1.2.3.1

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

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

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

Step 1.1.2.6.2

Multiply $2$ by $- 2$ .

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

Step 1.1.2.7

Multiply $2$ by $- 1$ .

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

Step 1.1.2.8

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

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

Step 1.1.2.9

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

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

Step 1.1.2.10

Subtract $4$ from $1$ .

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

Step 1.1.2.11

Multiply $- 2$ by $- 1$ .

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

Step 1.1.2.12

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

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

Step 1.1.2.13

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

$2 x + 2 x^{- 3}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

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

$2 x + 2 \frac{1}{x^{3}}$

Step 1.1.3.2

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

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.

$1, x^{3}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 2.3

Multiply each term in $2 x + \frac{2}{x^{3}} = 0$ by $x^{3}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $2 x + \frac{2}{x^{3}} = 0$ by $x^{3}$ .

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

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 $x^{3}$ by adding the exponents.

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Step 2.3.2.1.1.1

Move $x^{3}$ .

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

Step 2.3.2.1.1.2

Multiply $x^{3}$ by $x$ .

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Step 2.3.2.1.1.2.1

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

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

Step 2.3.2.1.1.2.2

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

$2 x^{3 + 1} + \frac{2}{x^{3}} x^{3} = 0 x^{3}$

Step 2.3.2.1.1.3

Add $3$ and $1$ .

$2 x^{4} + \frac{2}{x^{3}} x^{3} = 0 x^{3}$

Step 2.3.2.1.2

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

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Step 2.3.2.1.2.1

Cancel the common factor.

$2 x^{4} + \frac{2}{x^{3}} x^{3} = 0 x^{3}$

Step 2.3.2.1.2.2

Rewrite the expression.

$2 x^{4} + 2 = 0 x^{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0$ by $x^{3}$ .

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

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $2$ from both sides of the equation.

$2 x^{4} = - 2$

Step 2.4.2

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

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Step 2.4.2.1

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

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.1.2

Divide $x^{4}$ by $1$ .

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

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $- 2$ by $2$ .

$x^{4} = - 1$

Step 2.4.3

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

$x = \pm \sqrt[4]{- 1}$

Step 2.4.4

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

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Step 2.4.4.1

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

$x = \sqrt[4]{- 1}$

Step 2.4.4.2

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

$x = - \sqrt[4]{- 1}$

Step 2.4.4.3

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

$x = \sqrt[4]{- 1}, - \sqrt[4]{- 1}$

Step 2.5

Exclude the solutions that do not make $2 x + \frac{2}{x^{3}} = 0$ true.

$No solution$

Step 3

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 4

Find where the derivative is undefined.

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Step 4.1

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

$x^{3} = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

$x = 0$

Step 5

After finding the point that makes the derivative $f' (x) = 2 x + \frac{2}{x^{3}}$ equal to $0$ or undefined, the interval to check where $f (x) = x^{2} - \frac{1}{x^{2}}$ is increasing and where it is decreasing is $(- \infty, 0) \cup (0, \infty)$ .

$(- \infty, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $2$ by $- 1$ .

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

Step 6.2.1.2

Raise $- 1$ to the power of $3$ .

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

Step 6.2.1.3

Divide $2$ by $- 1$ .

$f' (- 1) = - 2 - 2$

Step 6.2.2

Subtract $2$ from $- 2$ .

$f' (- 1) = - 4$

Step 6.2.3

The final answer is $- 4$ .

$- 4$

Step 6.3

At $x = - 1$ the derivative is $- 4$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

Decreasing on $(- \infty, 0)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(0, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1$ in the expression.

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

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $2$ by $1$ .

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

Step 7.2.1.2

One to any power is one.

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

Step 7.2.1.3

Divide $2$ by $1$ .

$f' (1) = 2 + 2$

Step 7.2.2

Add $2$ and $2$ .

$f' (1) = 4$

Step 7.2.3

The final answer is $4$ .

$4$

Step 7.3

At $x = 1$ the derivative is $4$ . Since this is positive, the function is increasing on $(0, \infty)$ .

Increasing on $(0, \infty)$ since $f' (x) > 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(0, \infty)$

Decreasing on: $(- \infty, 0)$

Step 9