Calculus Examples

$g (x) = \frac{x^{2} + 4}{4 x}$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x^{2} + 4}{4 x}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{x^{2} + 4}{x}]$ .

$\frac{1}{4} \frac{d}{d x} [\frac{x^{2} + 4}{x}]$

Step 1.2

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 $f (x) = x^{2} + 4$ and $g (x) = x$ .

$\frac{1}{4} \cdot \frac{x \frac{d}{d x} [x^{2} + 4] - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

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

$\frac{1}{4} \cdot \frac{x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.2

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

$\frac{1}{4} \cdot \frac{x (2 x + \frac{d}{d x} [4]) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.3

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

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

Step 1.3.4

Add $2 x$ and $0$ .

$\frac{1}{4} \cdot \frac{x (2 x) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.4

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

$\frac{1}{4} \cdot \frac{2 (x^{1} x) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.5

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

$\frac{1}{4} \cdot \frac{2 (x^{1} x^{1}) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.6

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

$\frac{1}{4} \cdot \frac{2 x^{1 + 1} - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.7

Add $1$ and $1$ .

$\frac{1}{4} \cdot \frac{2 x^{2} - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 1.8

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

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

Step 1.9

Combine fractions.

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Step 1.9.1

Multiply $- 1$ by $1$ .

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

Step 1.9.2

Multiply $\frac{1}{4}$ by $\frac{2 x^{2} - (x^{2} + 4)}{x^{2}}$ .

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

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

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

Step 1.10.2

Simplify the numerator.

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Step 1.10.2.1

Multiply $- 1$ by $4$ .

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

Step 1.10.2.2

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

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

Step 1.10.3

Simplify the numerator.

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Step 1.10.3.1

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

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

Step 1.10.3.2

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 $b = 2$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{(x + 2) (x - 2)}{4 x^{2}}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{(x + 2) (x - 2)}{x^{2}}]$ .

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

Step 2.2

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $2$ by $2$ .

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

Step 2.4

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 $f (x) = x + 2$ and $g (x) = x - 2$ .

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

Step 2.5

Differentiate.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

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

Step 2.5.4.2

Multiply $x + 2$ by $1$ .

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

Step 2.5.5

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

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

Step 2.5.6

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

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

Step 2.5.7

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 2.5.8

Simplify by adding terms.

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Step 2.5.8.1

Add $1$ and $0$ .

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

Step 2.5.8.2

Multiply $x - 2$ by $1$ .

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

Step 2.5.8.3

Add $x$ and $x$ .

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

Step 2.5.8.4

Simplify by subtracting numbers.

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Step 2.5.8.4.1

Subtract $2$ from $2$ .

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

Step 2.5.8.4.2

Add $2 x$ and $0$ .

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

Step 2.6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.6.1

Move $x$ .

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

Step 2.6.2

Multiply $x$ by $x^{2}$ .

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Step 2.6.2.1

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

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

Step 2.6.2.2

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

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

Step 2.6.3

Add $1$ and $2$ .

$g'' (x) = \frac{1}{4} \cdot \frac{x^{3} \cdot 2 - (x + 2) (x - 2) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.7

Move $2$ to the left of $x^{3}$ .

$g'' (x) = \frac{1}{4} \cdot \frac{2 \cdot x^{3} - (x + 2) (x - 2) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.8

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

$g'' (x) = \frac{1}{4} \cdot \frac{2 x^{3} - (x + 2) (x - 2) (2 x)}{x^{4}}$

Step 2.9

Combine fractions.

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Step 2.9.1

Multiply $2$ by $- 1$ .

$g'' (x) = \frac{1}{4} \cdot \frac{2 x^{3} - 2 (x + 2) (x - 2) x}{x^{4}}$

Step 2.9.2

Multiply $\frac{1}{4}$ by $\frac{2 x^{3} - 2 (x + 2) (x - 2) x}{x^{4}}$ .

$g'' (x) = \frac{2 x^{3} - 2 (x + 2) (x - 2) x}{4 x^{4}}$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$g'' (x) = \frac{2 x^{3} + (- 2 x - 2 \cdot 2) (x - 2) x}{4 x^{4}}$

Step 2.10.2

Simplify the numerator.

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Step 2.10.2.1

Simplify each term.

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Step 2.10.2.1.1

Multiply $- 2$ by $2$ .

$g'' (x) = \frac{2 x^{3} + (- 2 x - 4) (x - 2) x}{4 x^{4}}$

Step 2.10.2.1.2

Expand $(- 2 x - 4) (x - 2)$ using the FOIL Method.

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Step 2.10.2.1.2.1

Apply the distributive property.

$g'' (x) = \frac{2 x^{3} + (- 2 x (x - 2) - 4 (x - 2)) x}{4 x^{4}}$

Step 2.10.2.1.2.2

Apply the distributive property.

$g'' (x) = \frac{2 x^{3} + (- 2 x \cdot x - 2 x \cdot - 2 - 4 (x - 2)) x}{4 x^{4}}$

Step 2.10.2.1.2.3

Apply the distributive property.

$g'' (x) = \frac{2 x^{3} + (- 2 x \cdot x - 2 x \cdot - 2 - 4 x - 4 \cdot - 2) x}{4 x^{4}}$

Step 2.10.2.1.3

Simplify and combine like terms.

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Step 2.10.2.1.3.1

Simplify each term.

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Step 2.10.2.1.3.1.1

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

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Step 2.10.2.1.3.1.1.1

Move $x$ .

$g'' (x) = \frac{2 x^{3} + (- 2 (x \cdot x) - 2 x \cdot - 2 - 4 x - 4 \cdot - 2) x}{4 x^{4}}$

Step 2.10.2.1.3.1.1.2

Multiply $x$ by $x$ .

$g'' (x) = \frac{2 x^{3} + (- 2 x^{2} - 2 x \cdot - 2 - 4 x - 4 \cdot - 2) x}{4 x^{4}}$

Step 2.10.2.1.3.1.2

Multiply $- 2$ by $- 2$ .

$g'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 4 x - 4 x - 4 \cdot - 2) x}{4 x^{4}}$

Step 2.10.2.1.3.1.3

Multiply $- 4$ by $- 2$ .

$g'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 4 x - 4 x + 8) x}{4 x^{4}}$

Step 2.10.2.1.3.2

Subtract $4 x$ from $4 x$ .

$g'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 0 + 8) x}{4 x^{4}}$

Step 2.10.2.1.3.3

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

$g'' (x) = \frac{2 x^{3} + (- 2 x^{2} + 8) x}{4 x^{4}}$

Step 2.10.2.1.4

Apply the distributive property.

$g'' (x) = \frac{2 x^{3} - 2 x^{2} x + 8 x}{4 x^{4}}$

Step 2.10.2.1.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.10.2.1.5.1

Move $x$ .

$g'' (x) = \frac{2 x^{3} - 2 (x \cdot x^{2}) + 8 x}{4 x^{4}}$

Step 2.10.2.1.5.2

Multiply $x$ by $x^{2}$ .

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Step 2.10.2.1.5.2.1

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

$g'' (x) = \frac{2 x^{3} - 2 (x \cdot x^{2}) + 8 x}{4 x^{4}}$

Step 2.10.2.1.5.2.2

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

$g'' (x) = \frac{2 x^{3} - 2 x^{1 + 2} + 8 x}{4 x^{4}}$

Step 2.10.2.1.5.3

Add $1$ and $2$ .

$g'' (x) = \frac{2 x^{3} - 2 x^{3} + 8 x}{4 x^{4}}$

Step 2.10.2.2

Subtract $2 x^{3}$ from $2 x^{3}$ .

$g'' (x) = \frac{0 + 8 x}{4 x^{4}}$

Step 2.10.2.3

Add $0$ and $8 x$ .

$g'' (x) = \frac{8 x}{4 x^{4}}$

Step 2.10.3

Combine terms.

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Step 2.10.3.1

Cancel the common factor of $8$ and $4$ .

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Step 2.10.3.1.1

Factor $4$ out of $8 x$ .

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

Step 2.10.3.1.2

Cancel the common factors.

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Step 2.10.3.1.2.1

Factor $4$ out of $4 x^{4}$ .

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

Step 2.10.3.1.2.2

Cancel the common factor.

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

Step 2.10.3.1.2.3

Rewrite the expression.

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

Step 2.10.3.2

Cancel the common factor of $x$ and $x^{4}$ .

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Step 2.10.3.2.1

Factor $x$ out of $2 x$ .

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

Step 2.10.3.2.2

Cancel the common factors.

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Step 2.10.3.2.2.1

Factor $x$ out of $x^{4}$ .

$g'' (x) = \frac{x \cdot 2}{x \cdot x^{3}}$

Step 2.10.3.2.2.2

Cancel the common factor.

$g'' (x) = \frac{x \cdot 2}{x \cdot x^{3}}$

Step 2.10.3.2.2.3

Rewrite the expression.

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

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x^{2} + 4}{4 x}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{x^{2} + 4}{x}]$ .

$\frac{1}{4} \frac{d}{d x} [\frac{x^{2} + 4}{x}]$

Step 4.1.2

$\frac{1}{4} \cdot \frac{x \frac{d}{d x} [x^{2} + 4] - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3

Differentiate.

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Step 4.1.3.1

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

$\frac{1}{4} \cdot \frac{x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.2

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

$\frac{1}{4} \cdot \frac{x (2 x + \frac{d}{d x} [4]) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.3

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

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

Step 4.1.3.4

Add $2 x$ and $0$ .

$\frac{1}{4} \cdot \frac{x (2 x) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.4

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

$\frac{1}{4} \cdot \frac{2 (x^{1} x) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.5

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

$\frac{1}{4} \cdot \frac{2 (x^{1} x^{1}) - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.6

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

$\frac{1}{4} \cdot \frac{2 x^{1 + 1} - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.7

Add $1$ and $1$ .

$\frac{1}{4} \cdot \frac{2 x^{2} - (x^{2} + 4) \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.8

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

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

Step 4.1.9

Combine fractions.

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Step 4.1.9.1

Multiply $- 1$ by $1$ .

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

Step 4.1.9.2

Multiply $\frac{1}{4}$ by $\frac{2 x^{2} - (x^{2} + 4)}{x^{2}}$ .

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

Step 4.1.10

Simplify.

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Step 4.1.10.1

Apply the distributive property.

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

Step 4.1.10.2

Simplify the numerator.

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Step 4.1.10.2.1

Multiply $- 1$ by $4$ .

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

Step 4.1.10.2.2

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

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

Step 4.1.10.3

Simplify the numerator.

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Step 4.1.10.3.1

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

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

Step 4.1.10.3.2

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 $b = 2$ .

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$(x + 2) (x - 2) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

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

$x + 2 = 0$

$x - 2 = 0$

Step 5.3.2

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 5.3.2.1

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

$x + 2 = 0$

Step 5.3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5.3.3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 5.3.3.1

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

$x - 2 = 0$

Step 5.3.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5.3.4

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

$x = - 2, 2$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$4 x^{2} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

Divide each term in $4 x^{2} = 0$ by $4$ and simplify.

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Step 6.2.1.1

Divide each term in $4 x^{2} = 0$ by $4$ .

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

Step 6.2.1.2

Simplify the left side.

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Step 6.2.1.2.1

Cancel the common factor of $4$ .

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Step 6.2.1.2.1.1

Cancel the common factor.

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

Step 6.2.1.2.1.2

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

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

Step 6.2.1.3

Simplify the right side.

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Step 6.2.1.3.1

Divide $0$ by $4$ .

$x^{2} = 0$

Step 6.2.2

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

$x = \pm \sqrt{0}$

Step 6.2.3

Simplify $\pm \sqrt{0}$ .

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

$x = \pm 0$

Step 6.2.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = - 2, 2$

Step 8

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2}{{(- 2)}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Cancel the common factor of $2$ and ${(- 2)}^{3}$ .

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Step 9.1.1

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- 1 (- 2)}{{(- 2)}^{3}}$

Step 9.1.2

Factor $- 2$ out of $- 1 (- 2)$ .

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

Step 9.1.3

Cancel the common factors.

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Step 9.1.3.1

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

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

Step 9.1.3.2

Cancel the common factor.

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

Step 9.1.3.3

Rewrite the expression.

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

Step 9.2

Simplify the expression.

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Step 9.2.1

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

$\frac{- 1}{4}$

Step 9.2.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 10

$x = - 2$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 2$ is a local maximum

Step 11

Find the y-value when $x = - 2$ .

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Step 11.1

Replace the variable $x$ with $- 2$ in the expression.

$g (- 2) = \frac{{(- 2)}^{2} + 4}{4 (- 2)}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify the numerator.

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Step 11.2.1.1

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

$g (- 2) = \frac{4 + 4}{4 (- 2)}$

Step 11.2.1.2

Add $4$ and $4$ .

$g (- 2) = \frac{8}{4 (- 2)}$

Step 11.2.2

Simplify the expression.

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Step 11.2.2.1

Multiply $4$ by $- 2$ .

$g (- 2) = \frac{8}{- 8}$

Step 11.2.2.2

Divide $8$ by $- 8$ .

$g (- 2) = - 1$

Step 11.2.3

The final answer is $- 1$ .

$y = - 1$

Step 12

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2}{{(2)}^{3}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Cancel the common factor of $2$ and ${(2)}^{3}$ .

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Step 13.1.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{{(2)}^{3}}$

Step 13.1.2

Cancel the common factors.

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Step 13.1.2.1

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

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

Step 13.1.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2^{2}}$

Step 13.1.2.3

Rewrite the expression.

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

Step 13.2

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

$\frac{1}{4}$

Step 14

$x = 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 2$ is a local minimum

Step 15

Find the y-value when $x = 2$ .

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Step 15.1

Replace the variable $x$ with $2$ in the expression.

$g (2) = \frac{{(2)}^{2} + 4}{4 (2)}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify the numerator.

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Step 15.2.1.1

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

$g (2) = \frac{4 + 4}{4 (2)}$

Step 15.2.1.2

Add $4$ and $4$ .

$g (2) = \frac{8}{4 (2)}$

Step 15.2.2

Simplify the expression.

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Step 15.2.2.1

Multiply $4$ by $2$ .

$g (2) = \frac{8}{8}$

Step 15.2.2.2

Divide $8$ by $8$ .

$g (2) = 1$

Step 15.2.3

The final answer is $1$ .

$y = 1$

Step 16

These are the local extrema for $g (x) = \frac{x^{2} + 4}{4 x}$ .

$(- 2, - 1)$ is a local maxima

$(2, 1)$ is a local minima

Step 17