Calculus Examples

$f (x) = \frac{x^{2} + 16}{2 x}$

Step 1

Find the first derivative.

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Step 1.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{2} + 16}{2 x}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{x^{2} + 16}{x}]$ .

$\frac{1}{2} \frac{d}{d x} [\frac{x^{2} + 16}{x}]$

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 $f (x) = x^{2} + 16$ and $g (x) = x$ .

$\frac{1}{2} \cdot \frac{x \frac{d}{d x} [x^{2} + 16] - (x^{2} + 16) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $x^{2} + 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [16]$ .

$\frac{1}{2} \cdot \frac{x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [16]) - (x^{2} + 16) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.2

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

$\frac{1}{2} \cdot \frac{x (2 x + \frac{d}{d x} [16]) - (x^{2} + 16) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.3

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$\frac{1}{2} \cdot \frac{x (2 x + 0) - (x^{2} + 16) \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.4

Add $2 x$ and $0$ .

$\frac{1}{2} \cdot \frac{x (2 x) - (x^{2} + 16) \frac{d}{d x} [x]}{x^{2}}$

Step 1.4

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

$\frac{1}{2} \cdot \frac{2 (x^{1} x) - (x^{2} + 16) \frac{d}{d x} [x]}{x^{2}}$

Step 1.5

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

$\frac{1}{2} \cdot \frac{2 (x^{1} x^{1}) - (x^{2} + 16) \frac{d}{d x} [x]}{x^{2}}$

Step 1.6

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

$\frac{1}{2} \cdot \frac{2 x^{1 + 1} - (x^{2} + 16) \frac{d}{d x} [x]}{x^{2}}$

Step 1.7

Add $1$ and $1$ .

$\frac{1}{2} \cdot \frac{2 x^{2} - (x^{2} + 16) \frac{d}{d x} [x]}{x^{2}}$

Step 1.8

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

$\frac{1}{2} \cdot \frac{2 x^{2} - (x^{2} + 16) \cdot 1}{x^{2}}$

Step 1.9

Combine fractions.

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Step 1.9.1

Multiply $- 1$ by $1$ .

$\frac{1}{2} \cdot \frac{2 x^{2} - (x^{2} + 16)}{x^{2}}$

Step 1.9.2

Multiply $\frac{1}{2}$ by $\frac{2 x^{2} - (x^{2} + 16)}{x^{2}}$ .

$\frac{2 x^{2} - (x^{2} + 16)}{2 x^{2}}$

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

$\frac{2 x^{2} - x^{2} - 1 \cdot 16}{2 x^{2}}$

Step 1.10.2

Simplify the numerator.

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Step 1.10.2.1

Multiply $- 1$ by $16$ .

$\frac{2 x^{2} - x^{2} - 16}{2 x^{2}}$

Step 1.10.2.2

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

$\frac{x^{2} - 16}{2 x^{2}}$

Step 1.10.3

Simplify the numerator.

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Step 1.10.3.1

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.2

Multiply $2$ by $2$ .

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

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 $f (x) = x + 4$ and $g (x) = x - 4$ .

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

Step 2.5

Differentiate.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

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

Step 2.5.4.2

Multiply $x + 4$ by $1$ .

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

Step 2.5.5

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

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

Step 2.5.6

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

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

Step 2.5.7

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

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

Step 2.5.8

Simplify by adding terms.

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Step 2.5.8.1

Add $1$ and $0$ .

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

Step 2.5.8.2

Multiply $x - 4$ by $1$ .

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

Step 2.5.8.3

Add $x$ and $x$ .

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

Step 2.5.8.4

Simplify by subtracting numbers.

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Step 2.5.8.4.1

Subtract $4$ from $4$ .

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

Step 2.5.8.4.2

Add $2 x$ and $0$ .

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

Step 2.6

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

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Step 2.6.1

Move $x$ .

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

Step 2.6.2

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

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Step 2.6.2.1

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

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

Step 2.6.2.2

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

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

Step 2.6.3

Add $1$ and $2$ .

$\frac{1}{2} \cdot \frac{x^{3} \cdot 2 - (x + 4) (x - 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.7

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

$\frac{1}{2} \cdot \frac{2 \cdot x^{3} - (x + 4) (x - 4) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.8

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

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

Step 2.9

Combine fractions.

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Step 2.9.1

Multiply $2$ by $- 1$ .

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

Step 2.9.2

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

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

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

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

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 $4$ .

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

Step 2.10.2.1.2

Expand $(- 2 x - 8) (x - 4)$ using the FOIL Method.

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Step 2.10.2.1.2.1

Apply the distributive property.

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

Step 2.10.2.1.2.2

Apply the distributive property.

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

Step 2.10.2.1.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 2.10.2.1.3.1.1.2

Multiply $x$ by $x$ .

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

Step 2.10.2.1.3.1.2

Multiply $- 4$ by $- 2$ .

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

Step 2.10.2.1.3.1.3

Multiply $- 8$ by $- 4$ .

$\frac{2 x^{3} + (- 2 x^{2} + 8 x - 8 x + 32) x}{2 x^{4}}$

Step 2.10.2.1.3.2

Subtract $8 x$ from $8 x$ .

$\frac{2 x^{3} + (- 2 x^{2} + 0 + 32) x}{2 x^{4}}$

Step 2.10.2.1.3.3

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

$\frac{2 x^{3} + (- 2 x^{2} + 32) x}{2 x^{4}}$

Step 2.10.2.1.4

Apply the distributive property.

$\frac{2 x^{3} - 2 x^{2} x + 32 x}{2 x^{4}}$

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 $x$ .

$\frac{2 x^{3} - 2 (x \cdot x^{2}) + 32 x}{2 x^{4}}$

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 $1$ .

$\frac{2 x^{3} - 2 (x^{1} x^{2}) + 32 x}{2 x^{4}}$

Step 2.10.2.1.5.2.2

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

$\frac{2 x^{3} - 2 x^{1 + 2} + 32 x}{2 x^{4}}$

Step 2.10.2.1.5.3

Add $1$ and $2$ .

$\frac{2 x^{3} - 2 x^{3} + 32 x}{2 x^{4}}$

Step 2.10.2.2

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

$\frac{0 + 32 x}{2 x^{4}}$

Step 2.10.2.3

Add $0$ and $32 x$ .

$\frac{32 x}{2 x^{4}}$

Step 2.10.3

Combine terms.

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Step 2.10.3.1

Cancel the common factor of $32$ and $2$ .

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Step 2.10.3.1.1

Factor $2$ out of $32 x$ .

$\frac{2 (16 x)}{2 x^{4}}$

Step 2.10.3.1.2

Cancel the common factors.

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Step 2.10.3.1.2.1

Factor $2$ out of $2 x^{4}$ .

$\frac{2 (16 x)}{2 (x^{4})}$

Step 2.10.3.1.2.2

Cancel the common factor.

$\frac{2 (16 x)}{2 x^{4}}$

Step 2.10.3.1.2.3

Rewrite the expression.

$\frac{16 x}{x^{4}}$

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 $16 x$ .

$\frac{x \cdot 16}{x^{4}}$

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 $x^{4}$ .

$\frac{x \cdot 16}{x \cdot x^{3}}$

Step 2.10.3.2.2.2

Cancel the common factor.

$\frac{x \cdot 16}{x \cdot x^{3}}$

Step 2.10.3.2.2.3

Rewrite the expression.

$f'' (x) = \frac{16}{x^{3}}$

Step 3

Find the third derivative.

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Step 3.1

Since $16$ is constant with respect to $x$ , the derivative of $\frac{16}{x^{3}}$ with respect to $x$ is $16 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$16 \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$16 \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 3.2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.2.2.1

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

$16 \frac{d}{d x} [x^{3 \cdot - 1}]$

Step 3.2.2.2

Multiply $3$ by $- 1$ .

$16 \frac{d}{d x} [x^{- 3}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$16 (- 3 x^{- 4})$

Step 3.4

Multiply $- 3$ by $16$ .

$- 48 x^{- 4}$

Step 3.5

Simplify.

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Step 3.5.1

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

$- 48 \frac{1}{x^{4}}$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Combine $- 48$ and $\frac{1}{x^{4}}$ .

$\frac{- 48}{x^{4}}$

Step 3.5.2.2

Move the negative in front of the fraction.

$f''' (x) = - \frac{48}{x^{4}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- \frac{48}{x^{4}}$ with respect to $x$ is $- 48 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$- 48 \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$- 48 \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 4.2.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 4.2.2.1

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

$- 48 \frac{d}{d x} [x^{4 \cdot - 1}]$

Step 4.2.2.2

Multiply $4$ by $- 1$ .

$- 48 \frac{d}{d x} [x^{- 4}]$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 4$ .

$- 48 (- 4 x^{- 5})$

Step 4.4

Multiply $- 4$ by $- 48$ .

$192 x^{- 5}$

Step 4.5

Simplify.

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Step 4.5.1

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

$192 \frac{1}{x^{5}}$

Step 4.5.2

Combine $192$ and $\frac{1}{x^{5}}$ .

$f^{4} (x) = \frac{192}{x^{5}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $\frac{192}{x^{5}}$ .

$\frac{192}{x^{5}}$