Calculus Examples

$y = \frac{2 x - 3}{2 x - 8}$

Step 1

Find the first derivative.

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Step 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 $f (x) = 2 x - 3$ and $g (x) = 2 x - 8$ .

$\frac{(2 x - 8) \frac{d}{d x} [2 x - 3] - (2 x - 3) \frac{d}{d x} [2 x - 8]}{{(2 x - 8)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $2 x - 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$ .

$\frac{(2 x - 8) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]) - (2 x - 3) \frac{d}{d x} [2 x - 8]}{{(2 x - 8)}^{2}}$

Step 1.2.2

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

$\frac{(2 x - 8) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]) - (2 x - 3) \frac{d}{d x} [2 x - 8]}{{(2 x - 8)}^{2}}$

Step 1.2.3

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

$\frac{(2 x - 8) (2 \cdot 1 + \frac{d}{d x} [- 3]) - (2 x - 3) \frac{d}{d x} [2 x - 8]}{{(2 x - 8)}^{2}}$

Step 1.2.4

Multiply $2$ by $1$ .

$\frac{(2 x - 8) (2 + \frac{d}{d x} [- 3]) - (2 x - 3) \frac{d}{d x} [2 x - 8]}{{(2 x - 8)}^{2}}$

Step 1.2.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{(2 x - 8) (2 + 0) - (2 x - 3) \frac{d}{d x} [2 x - 8]}{{(2 x - 8)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2$ and $0$ .

$\frac{(2 x - 8) \cdot 2 - (2 x - 3) \frac{d}{d x} [2 x - 8]}{{(2 x - 8)}^{2}}$

Step 1.2.6.2

Move $2$ to the left of $2 x - 8$ .

$\frac{2 \cdot (2 x - 8) - (2 x - 3) \frac{d}{d x} [2 x - 8]}{{(2 x - 8)}^{2}}$

Step 1.2.7

By the Sum Rule, the derivative of $2 x - 8$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 8]$ .

$\frac{2 (2 x - 8) - (2 x - 3) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 8])}{{(2 x - 8)}^{2}}$

Step 1.2.8

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

$\frac{2 (2 x - 8) - (2 x - 3) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 8])}{{(2 x - 8)}^{2}}$

Step 1.2.9

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

$\frac{2 (2 x - 8) - (2 x - 3) (2 \cdot 1 + \frac{d}{d x} [- 8])}{{(2 x - 8)}^{2}}$

Step 1.2.10

Multiply $2$ by $1$ .

$\frac{2 (2 x - 8) - (2 x - 3) (2 + \frac{d}{d x} [- 8])}{{(2 x - 8)}^{2}}$

Step 1.2.11

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$\frac{2 (2 x - 8) - (2 x - 3) (2 + 0)}{{(2 x - 8)}^{2}}$

Step 1.2.12

Simplify the expression.

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Step 1.2.12.1

Add $2$ and $0$ .

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

Step 1.2.12.2

Multiply $2$ by $- 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Combine the opposite terms in $2 (2 x) + 2 \cdot - 8 - 2 (2 x) - 2 \cdot - 3$ .

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Step 1.3.3.1.1

Subtract $2 (2 x)$ from $2 (2 x)$ .

$\frac{2 \cdot - 8 + 0 - 2 \cdot - 3}{{(2 x - 8)}^{2}}$

Step 1.3.3.1.2

Add $2 \cdot - 8$ and $0$ .

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

Step 1.3.3.2

Simplify each term.

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Step 1.3.3.2.1

Multiply $2$ by $- 8$ .

$\frac{- 16 - 2 \cdot - 3}{{(2 x - 8)}^{2}}$

Step 1.3.3.2.2

Multiply $- 2$ by $- 3$ .

$\frac{- 16 + 6}{{(2 x - 8)}^{2}}$

Step 1.3.3.3

Add $- 16$ and $6$ .

$\frac{- 10}{{(2 x - 8)}^{2}}$

Step 1.3.4

Move the negative in front of the fraction.

$- \frac{10}{{(2 x - 8)}^{2}}$

Step 1.3.5

Simplify the denominator.

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Step 1.3.5.1

Factor $2$ out of $2 x - 8$ .

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Step 1.3.5.1.1

Factor $2$ out of $2 x$ .

$- \frac{10}{{(2 (x) - 8)}^{2}}$

Step 1.3.5.1.2

Factor $2$ out of $- 8$ .

$- \frac{10}{{(2 x + 2 \cdot - 4)}^{2}}$

Step 1.3.5.1.3

Factor $2$ out of $2 x + 2 \cdot - 4$ .

$- \frac{10}{{(2 (x - 4))}^{2}}$

Step 1.3.5.2

Apply the product rule to $2 (x - 4)$ .

$- \frac{10}{2^{2} {(x - 4)}^{2}}$

Step 1.3.5.3

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

$- \frac{10}{4 {(x - 4)}^{2}}$

Step 1.3.6

Cancel the common factor of $10$ and $4$ .

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Step 1.3.6.1

Factor $2$ out of $10$ .

$- \frac{2 (5)}{4 {(x - 4)}^{2}}$

Step 1.3.6.2

Cancel the common factors.

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Step 1.3.6.2.1

Factor $2$ out of $4 {(x - 4)}^{2}$ .

$- \frac{2 (5)}{2 (2 {(x - 4)}^{2})}$

Step 1.3.6.2.2

Cancel the common factor.

$- \frac{2 \cdot 5}{2 (2 {(x - 4)}^{2})}$

Step 1.3.6.2.3

Rewrite the expression.

$f' (x) = - \frac{5}{2 {(x - 4)}^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Since $- \frac{5}{2}$ is constant with respect to $x$ , the derivative of $- \frac{5}{2 {(x - 4)}^{2}}$ with respect to $x$ is $- \frac{5}{2} \frac{d}{d x} [\frac{1}{{(x - 4)}^{2}}]$ .

$- \frac{5}{2} \frac{d}{d x} [\frac{1}{{(x - 4)}^{2}}]$

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite $\frac{1}{{(x - 4)}^{2}}$ as ${({(x - 4)}^{2})}^{- 1}$ .

$- \frac{5}{2} \frac{d}{d x} [{({(x - 4)}^{2})}^{- 1}]$

Step 2.1.2.2

Multiply the exponents in ${({(x - 4)}^{2})}^{- 1}$ .

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Step 2.1.2.2.1

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

$- \frac{5}{2} \frac{d}{d x} [{(x - 4)}^{2 \cdot - 1}]$

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

$- \frac{5}{2} \frac{d}{d x} [{(x - 4)}^{- 2}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 2}$ and $g (x) = x - 4$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x - 4$ .

$- \frac{5}{2} (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [x - 4])$

Step 2.2.2

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

$- \frac{5}{2} (- 2 u^{- 3} \frac{d}{d x} [x - 4])$

Step 2.2.3

Replace all occurrences of $u$ with $x - 4$ .

$- \frac{5}{2} (- 2 {(x - 4)}^{- 3} \frac{d}{d x} [x - 4])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $- 1$ .

$2 (\frac{5}{2}) ({(x - 4)}^{- 3} \frac{d}{d x} [x - 4])$

Step 2.3.2

Simplify terms.

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Step 2.3.2.1

Combine $2$ and $\frac{5}{2}$ .

$\frac{2 \cdot 5}{2} ({(x - 4)}^{- 3} \frac{d}{d x} [x - 4])$

Step 2.3.2.2

Multiply $2$ by $5$ .

$\frac{10}{2} ({(x - 4)}^{- 3} \frac{d}{d x} [x - 4])$

Step 2.3.2.3

Combine ${(x - 4)}^{- 3}$ and $\frac{10}{2}$ .

$\frac{{(x - 4)}^{- 3} \cdot 10}{2} \frac{d}{d x} [x - 4]$

Step 2.3.2.4

Simplify the expression.

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Step 2.3.2.4.1

Move $10$ to the left of ${(x - 4)}^{- 3}$ .

$\frac{10 \cdot {(x - 4)}^{- 3}}{2} \frac{d}{d x} [x - 4]$

Step 2.3.2.4.2

Move ${(x - 4)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{10}{2 {(x - 4)}^{3}} \frac{d}{d x} [x - 4]$

Step 2.3.2.5

Cancel the common factor of $10$ and $2$ .

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Step 2.3.2.5.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2 {(x - 4)}^{3}} \frac{d}{d x} [x - 4]$

Step 2.3.2.5.2

Cancel the common factors.

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Step 2.3.2.5.2.1

Factor $2$ out of $2 {(x - 4)}^{3}$ .

$\frac{2 \cdot 5}{2 ({(x - 4)}^{3})} \frac{d}{d x} [x - 4]$

Step 2.3.2.5.2.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 {(x - 4)}^{3}} \frac{d}{d x} [x - 4]$

Step 2.3.2.5.2.3

Rewrite the expression.

$\frac{5}{{(x - 4)}^{3}} \frac{d}{d x} [x - 4]$

Step 2.3.3

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

$\frac{5}{{(x - 4)}^{3}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 2.3.4

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

$\frac{5}{{(x - 4)}^{3}} (1 + \frac{d}{d x} [- 4])$

Step 2.3.5

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

$\frac{5}{{(x - 4)}^{3}} (1 + 0)$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $1$ and $0$ .

$\frac{5}{{(x - 4)}^{3}} \cdot 1$

Step 2.3.6.2

Multiply $\frac{5}{{(x - 4)}^{3}}$ by $1$ .

$f'' (x) = \frac{5}{{(x - 4)}^{3}}$

Step 3

Find the third derivative.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{{(x - 4)}^{3}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{{(x - 4)}^{3}}]$ .

$5 \frac{d}{d x} [\frac{1}{{(x - 4)}^{3}}]$

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(x - 4)}^{3}}$ as ${({(x - 4)}^{3})}^{- 1}$ .

$5 \frac{d}{d x} [{({(x - 4)}^{3})}^{- 1}]$

Step 3.1.2.2

Multiply the exponents in ${({(x - 4)}^{3})}^{- 1}$ .

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Step 3.1.2.2.1

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

$5 \frac{d}{d x} [{(x - 4)}^{3 \cdot - 1}]$

Step 3.1.2.2.2

Multiply $3$ by $- 1$ .

$5 \frac{d}{d x} [{(x - 4)}^{- 3}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 3}$ and $g (x) = x - 4$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x - 4$ .

$5 (\frac{d}{d u} [u^{- 3}] \frac{d}{d x} [x - 4])$

Step 3.2.2

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

$5 (- 3 u^{- 4} \frac{d}{d x} [x - 4])$

Step 3.2.3

Replace all occurrences of $u$ with $x - 4$ .

$5 (- 3 {(x - 4)}^{- 4} \frac{d}{d x} [x - 4])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $- 3$ by $5$ .

$- 15 ({(x - 4)}^{- 4} \frac{d}{d x} [x - 4])$

Step 3.3.2

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

$- 15 {(x - 4)}^{- 4} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 3.3.3

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

$- 15 {(x - 4)}^{- 4} (1 + \frac{d}{d x} [- 4])$

Step 3.3.4

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

$- 15 {(x - 4)}^{- 4} (1 + 0)$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $1$ and $0$ .

$- 15 {(x - 4)}^{- 4} \cdot 1$

Step 3.3.5.2

Multiply $- 15$ by $1$ .

$- 15 {(x - 4)}^{- 4}$

Step 3.4

Simplify.

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Step 3.4.1

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

$- 15 \frac{1}{{(x - 4)}^{4}}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 15$ and $\frac{1}{{(x - 4)}^{4}}$ .

$\frac{- 15}{{(x - 4)}^{4}}$

Step 3.4.2.2

Move the negative in front of the fraction.

$f''' (x) = - \frac{15}{{(x - 4)}^{4}}$

Step 4

Find the fourth derivative.

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Step 4.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- \frac{15}{{(x - 4)}^{4}}$ with respect to $x$ is $- 15 \frac{d}{d x} [\frac{1}{{(x - 4)}^{4}}]$ .

$- 15 \frac{d}{d x} [\frac{1}{{(x - 4)}^{4}}]$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

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

$- 15 \frac{d}{d x} [{({(x - 4)}^{4})}^{- 1}]$

Step 4.1.2.2

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

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Step 4.1.2.2.1

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

$- 15 \frac{d}{d x} [{(x - 4)}^{4 \cdot - 1}]$

Step 4.1.2.2.2

Multiply $4$ by $- 1$ .

$- 15 \frac{d}{d x} [{(x - 4)}^{- 4}]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 4}$ and $g (x) = x - 4$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $x - 4$ .

$- 15 (\frac{d}{d u} [u^{- 4}] \frac{d}{d x} [x - 4])$

Step 4.2.2

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

$- 15 (- 4 u^{- 5} \frac{d}{d x} [x - 4])$

Step 4.2.3

Replace all occurrences of $u$ with $x - 4$ .

$- 15 (- 4 {(x - 4)}^{- 5} \frac{d}{d x} [x - 4])$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply $- 4$ by $- 15$ .

$60 ({(x - 4)}^{- 5} \frac{d}{d x} [x - 4])$

Step 4.3.2

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

$60 {(x - 4)}^{- 5} (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 4.3.3

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

$60 {(x - 4)}^{- 5} (1 + \frac{d}{d x} [- 4])$

Step 4.3.4

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

$60 {(x - 4)}^{- 5} (1 + 0)$

Step 4.3.5

Simplify the expression.

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Step 4.3.5.1

Add $1$ and $0$ .

$60 {(x - 4)}^{- 5} \cdot 1$

Step 4.3.5.2

Multiply $60$ by $1$ .

$60 {(x - 4)}^{- 5}$

Step 4.4

Simplify.

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Step 4.4.1

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

$60 \frac{1}{{(x - 4)}^{5}}$

Step 4.4.2

Combine $60$ and $\frac{1}{{(x - 4)}^{5}}$ .

$f^{4} (x) = \frac{60}{{(x - 4)}^{5}}$