Calculus Examples

$\sqrt{4 x - 8}$

Step 1

Find the first derivative.

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Step 1.1

Rewrite $\frac{d}{d x} [\sqrt{4 x - 8}]$ as $\frac{d}{d x} [2 \sqrt{x - 2}]$ .

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Step 1.1.1

Rewrite $4 x - 8$ as $2^{2} (x - 2)$ .

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Step 1.1.1.1

Factor $4$ out of $4 x$ .

$\frac{d}{d x} [\sqrt{4 (x) - 8}]$

Step 1.1.1.2

Factor $4$ out of $- 8$ .

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

Step 1.1.1.3

Factor $4$ out of $4 (x) + 4 (- 2)$ .

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

Step 1.1.1.4

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

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

Step 1.1.2

Pull terms out from under the radical.

$\frac{d}{d x} [2 \sqrt{x - 2}]$

Step 1.2

Differentiate using the Constant Multiple Rule.

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 2}$ as ${(x - 2)}^{\frac{1}{2}}$ .

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

Step 1.2.2

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

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

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x - 2$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 1.3.2

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

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

Step 1.3.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.5

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Simplify terms.

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Step 1.8.1

Move the negative in front of the fraction.

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

Step 1.8.2

Combine $\frac{1}{2}$ and ${(x - 2)}^{- \frac{1}{2}}$ .

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

Step 1.8.3

Move ${(x - 2)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.8.4

Combine $\frac{1}{2 {(x - 2)}^{\frac{1}{2}}}$ and $2$ .

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

Step 1.8.5

Cancel the common factor.

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

Step 1.8.6

Rewrite the expression.

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

Simplify the expression.

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Step 1.12.1

Add $1$ and $0$ .

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

Step 1.12.2

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

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

Step 2

Find the second derivative.

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Step 2.1

Apply basic rules of exponents.

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Step 2.1.1

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

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 2.1.2.3

Move the negative in front of the fraction.

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

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $- 1$ .

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

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

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

Step 2.7.2

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

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

Step 2.7.3

Move ${(x - 2)}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

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

Step 2.11.2

Multiply $- 1$ by $1$ .

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

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 $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2 {(x - 2)}^{\frac{3}{2}}}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [\frac{1}{{(x - 2)}^{\frac{3}{2}}}]$ .

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

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(x - 2)}^{\frac{3}{2}}}$ as ${({(x - 2)}^{\frac{3}{2}})}^{- 1}$ .

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

Step 3.1.2.2

Multiply the exponents in ${({(x - 2)}^{\frac{3}{2}})}^{- 1}$ .

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.1.2.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 3.1.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.1.2.2.3

Move the negative in front of the fraction.

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

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 $f (x) = x^{- \frac{3}{2}}$ and $g (x) = x - 2$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{3}{2}$ .

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

Step 3.2.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.6.2

Subtract $2$ from $- 3$ .

$- \frac{1}{2} (- \frac{3}{2} {(x - 2)}^{\frac{- 5}{2}} \frac{d}{d x} [x - 2])$

Step 3.7

Combine fractions.

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Step 3.7.1

Move the negative in front of the fraction.

$- \frac{1}{2} (- \frac{3}{2} {(x - 2)}^{- \frac{5}{2}} \frac{d}{d x} [x - 2])$

Step 3.7.2

Combine ${(x - 2)}^{- \frac{5}{2}}$ and $\frac{3}{2}$ .

$- \frac{1}{2} (- \frac{{(x - 2)}^{- \frac{5}{2}} \cdot 3}{2} \frac{d}{d x} [x - 2])$

Step 3.7.3

Simplify the expression.

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Step 3.7.3.1

Move $3$ to the left of ${(x - 2)}^{- \frac{5}{2}}$ .

$- \frac{1}{2} (- \frac{3 \cdot {(x - 2)}^{- \frac{5}{2}}}{2} \frac{d}{d x} [x - 2])$

Step 3.7.3.2

Move ${(x - 2)}^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{2} (- \frac{3}{2 {(x - 2)}^{\frac{5}{2}}} \frac{d}{d x} [x - 2])$

Step 3.7.3.3

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) (\frac{3}{2 {(x - 2)}^{\frac{5}{2}}} \frac{d}{d x} [x - 2])$

Step 3.7.3.4

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} (\frac{3}{2 {(x - 2)}^{\frac{5}{2}}} \frac{d}{d x} [x - 2])$

Step 3.7.4

Multiply $\frac{3}{2 {(x - 2)}^{\frac{5}{2}}}$ by $\frac{1}{2}$ .

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

Step 3.7.5

Multiply $2$ by $2$ .

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

Step 3.8

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

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

Step 3.9

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

$\frac{3}{4 {(x - 2)}^{\frac{5}{2}}} (1 + \frac{d}{d x} [- 2])$

Step 3.10

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

$\frac{3}{4 {(x - 2)}^{\frac{5}{2}}} (1 + 0)$

Step 3.11

Simplify the expression.

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Step 3.11.1

Add $1$ and $0$ .

$\frac{3}{4 {(x - 2)}^{\frac{5}{2}}} \cdot 1$

Step 3.11.2

Multiply $\frac{3}{4 {(x - 2)}^{\frac{5}{2}}}$ by $1$ .

$f''' (x) = \frac{3}{4 {(x - 2)}^{\frac{5}{2}}}$

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 $\frac{3}{4}$ is constant with respect to $x$ , the derivative of $\frac{3}{4 {(x - 2)}^{\frac{5}{2}}}$ with respect to $x$ is $\frac{3}{4} \frac{d}{d x} [\frac{1}{{(x - 2)}^{\frac{5}{2}}}]$ .

$\frac{3}{4} \frac{d}{d x} [\frac{1}{{(x - 2)}^{\frac{5}{2}}}]$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

Rewrite $\frac{1}{{(x - 2)}^{\frac{5}{2}}}$ as ${({(x - 2)}^{\frac{5}{2}})}^{- 1}$ .

$\frac{3}{4} \frac{d}{d x} [{({(x - 2)}^{\frac{5}{2}})}^{- 1}]$

Step 4.1.2.2

Multiply the exponents in ${({(x - 2)}^{\frac{5}{2}})}^{- 1}$ .

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Step 4.1.2.2.1

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

$\frac{3}{4} \frac{d}{d x} [{(x - 2)}^{\frac{5}{2} \cdot - 1}]$

Step 4.1.2.2.2

Multiply $\frac{5}{2} \cdot - 1$ .

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Step 4.1.2.2.2.1

Combine $\frac{5}{2}$ and $- 1$ .

$\frac{3}{4} \frac{d}{d x} [{(x - 2)}^{\frac{5 \cdot - 1}{2}}]$

Step 4.1.2.2.2.2

Multiply $5$ by $- 1$ .

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

Step 4.1.2.2.3

Move the negative in front of the fraction.

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

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 $f (x) = x^{- \frac{5}{2}}$ and $g (x) = x - 2$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{5}{2}$ .

$\frac{3}{4} (- \frac{5}{2} u^{- \frac{5}{2} - 1} \frac{d}{d x} [x - 2])$

Step 4.2.3

Replace all occurrences of $u$ with $x - 2$ .

$\frac{3}{4} (- \frac{5}{2} {(x - 2)}^{- \frac{5}{2} - 1} \frac{d}{d x} [x - 2])$

Step 4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{4} (- \frac{5}{2} {(x - 2)}^{- \frac{5}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x - 2])$

Step 4.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{3}{4} (- \frac{5}{2} {(x - 2)}^{- \frac{5}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x - 2])$

Step 4.5

Combine the numerators over the common denominator.

$\frac{3}{4} (- \frac{5}{2} {(x - 2)}^{\frac{- 5 - 1 \cdot 2}{2}} \frac{d}{d x} [x - 2])$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $- 5$ .

$\frac{3}{4} (- \frac{5}{2} {(x - 2)}^{\frac{- 7}{2}} \frac{d}{d x} [x - 2])$

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

$\frac{3}{4} (- \frac{5}{2} {(x - 2)}^{- \frac{7}{2}} \frac{d}{d x} [x - 2])$

Step 4.7.2

Combine ${(x - 2)}^{- \frac{7}{2}}$ and $\frac{5}{2}$ .

$\frac{3}{4} (- \frac{{(x - 2)}^{- \frac{7}{2}} \cdot 5}{2} \frac{d}{d x} [x - 2])$

Step 4.7.3

Simplify the expression.

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Step 4.7.3.1

Move $5$ to the left of ${(x - 2)}^{- \frac{7}{2}}$ .

$\frac{3}{4} (- \frac{5 \cdot {(x - 2)}^{- \frac{7}{2}}}{2} \frac{d}{d x} [x - 2])$

Step 4.7.3.2

Move ${(x - 2)}^{- \frac{7}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{4} (- \frac{5}{2 {(x - 2)}^{\frac{7}{2}}} \frac{d}{d x} [x - 2])$

Step 4.7.4

Multiply $\frac{3}{4}$ by $\frac{5}{2 {(x - 2)}^{\frac{7}{2}}}$ .

$- \frac{3 \cdot 5}{4 (2 {(x - 2)}^{\frac{7}{2}})} \frac{d}{d x} [x - 2]$

Step 4.7.5

Multiply.

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Step 4.7.5.1

Multiply $3$ by $5$ .

$- \frac{15}{4 (2 {(x - 2)}^{\frac{7}{2}})} \frac{d}{d x} [x - 2]$

Step 4.7.5.2

Multiply $2$ by $4$ .

$- \frac{15}{8 {(x - 2)}^{\frac{7}{2}}} \frac{d}{d x} [x - 2]$

Step 4.8

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

$- \frac{15}{8 {(x - 2)}^{\frac{7}{2}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2])$

Step 4.9

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

$- \frac{15}{8 {(x - 2)}^{\frac{7}{2}}} (1 + \frac{d}{d x} [- 2])$

Step 4.10

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

$- \frac{15}{8 {(x - 2)}^{\frac{7}{2}}} (1 + 0)$

Step 4.11

Simplify the expression.

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Step 4.11.1

Add $1$ and $0$ .

$- \frac{15}{8 {(x - 2)}^{\frac{7}{2}}} \cdot 1$

Step 4.11.2

Multiply $- 1$ by $1$ .

$f^{4} (x) = - \frac{15}{8 {(x - 2)}^{\frac{7}{2}}}$