Calculus Examples

$f (x) = \frac{x}{x^{2} - 4}$

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 using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = x^{2} - 4$ .

$f' (x) = \frac{(x^{2} - 4) \frac{d}{d x} (x) - x \frac{d}{d x} (x^{2} - 4)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Multiply $x^{2} - 4$ by $1$ .

$f' (x) = \frac{x^{2} - 4 - x \frac{d}{d x} (x^{2} - 4)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.3

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

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

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 $n = 2$ .

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

Step 1.1.2.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{x^{2} - 4 - x (2 x)}{{(x^{2} - 4)}^{2}}$

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Add $1$ and $1$ .

$f' (x) = \frac{x^{2} - 4 - 2 x^{2}}{{(x^{2} - 4)}^{2}}$

Step 1.1.7

Subtract $2 x^{2}$ from $x^{2}$ .

$f' (x) = \frac{- x^{2} - 4}{{(x^{2} - 4)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{- x^{2} - 4}{{(x^{2} - 4)}^{2}}$ .

$\frac{- x^{2} - 4}{{(x^{2} - 4)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{- x^{2} - 4}{{(x^{2} - 4)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{- x^{2} - 4}{{(x^{2} - 4)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

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

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Add $4$ to both sides of the equation.

$- x^{2} = 4$

Step 2.3.2

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

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Step 2.3.2.1

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

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

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2.2

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

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

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $4$ by $- 1$ .

$x^{2} = - 4$

Step 2.3.3

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

$x = \pm \sqrt{- 4}$

Step 2.3.4

Simplify $\pm \sqrt{- 4}$ .

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Step 2.3.4.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 2.3.4.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.3.4.3

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

$x = \pm i \cdot \sqrt{4}$

Step 2.3.4.4

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

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

Step 2.3.4.5

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

$x = \pm i \cdot 2$

Step 2.3.4.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 2.3.5

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

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Step 2.3.5.1

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

$x = 2 i$

Step 2.3.5.2

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

$x = - 2 i$

Step 2.3.5.3

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

$x = 2 i, - 2 i$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{- x^{2} - 4}{{(x^{2} - 4)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x^{2} - 4)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Factor the left side of the equation.

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Apply the product rule to $(x + 2) (x - 2)$ .

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

Step 3.2.2

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

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

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

Step 3.2.3

Set ${(x + 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set ${(x + 2)}^{2}$ equal to $0$ .

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

Step 3.2.3.2

Solve ${(x + 2)}^{2} = 0$ for $x$ .

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Step 3.2.3.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.2.3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.2.4

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.4.1

Set ${(x - 2)}^{2}$ equal to $0$ .

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

Step 3.2.4.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 3.2.4.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.2.4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.2.5

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

$x = - 2, 2$

Step 3.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = 2$

Step 4

Evaluate $\frac{x}{x^{2} - 4}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 2$ .

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Step 4.1.1

Substitute $- 2$ for $x$ .

$\frac{- 2}{{(- 2)}^{2} - 4}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$\frac{- 2}{4 - 4}$

Step 4.1.2.2

Subtract $4$ from $4$ .

$\frac{- 2}{0}$

Step 4.1.2.3

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

Undefined

Step 4.2

Evaluate at $x = 2$ .

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Step 4.2.1

Substitute $2$ for $x$ .

$\frac{2}{{(2)}^{2} - 4}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$\frac{2}{4 - 4}$

Step 4.2.2.2

Subtract $4$ from $4$ .

$\frac{2}{0}$

Step 4.2.2.3

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

Undefined

Step 5

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

No critical points found