Calculus Examples

$\frac{2 x}{\sqrt{3 x - 1}}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 1}$ as ${(3 x - 1)}^{\frac{1}{2}}$ .

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

Step 1.1.2

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

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

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 $f (x) = x$ and $g (x) = {(3 x - 1)}^{\frac{1}{2}}$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.4

Simplify.

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

Step 1.5

Differentiate using the Power Rule.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.6

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 $g (x) = 3 x - 1$ .

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Step 1.6.1

To apply the Chain Rule, set $u$ as $3 x - 1$ .

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

Step 1.6.2

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

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

Step 1.6.3

Replace all occurrences of $u$ with $3 x - 1$ .

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Combine the numerators over the common denominator.

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

Step 1.10

Simplify the numerator.

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Step 1.10.1

Multiply $- 1$ by $2$ .

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

Step 1.10.2

Subtract $2$ from $1$ .

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

Step 1.11

Combine fractions.

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Step 1.11.1

Move the negative in front of the fraction.

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

Step 1.11.2

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

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

Step 1.11.3

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

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

Step 1.11.4

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

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

Step 1.12

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

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

Step 1.13

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

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

Step 1.14

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

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

Step 1.15

Multiply $3$ by $1$ .

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

Step 1.16

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 1.17

Combine fractions.

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Step 1.17.1

Add $3$ and $0$ .

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

Step 1.17.2

Multiply $3$ by $- 1$ .

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

Step 1.17.3

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

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

Step 1.17.4

Move the negative in front of the fraction.

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

Step 1.18

To write ${(3 x - 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(3 x - 1)}^{\frac{1}{2}}}{2 {(3 x - 1)}^{\frac{1}{2}}}$ .

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

Step 1.19

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

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

Step 1.20

Combine the numerators over the common denominator.

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

Step 1.21

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

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Step 1.21.1

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

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

Step 1.21.2

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

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

Step 1.21.3

Combine the numerators over the common denominator.

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

Step 1.21.4

Add $1$ and $1$ .

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

Step 1.21.5

Divide $2$ by $2$ .

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

Step 1.22

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

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

Step 1.23

Move $2$ to the left of $3 x - 1$ .

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

Step 1.24

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

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

Step 1.25

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

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

Step 1.26

Raise $3 x - 1$ to the power of $1$ .

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

Step 1.27

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

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

Step 1.28

Simplify the expression.

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Step 1.28.1

Write $1$ as a fraction with a common denominator.

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

Step 1.28.2

Combine the numerators over the common denominator.

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

Step 1.28.3

Add $2$ and $1$ .

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

Step 1.29

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

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

Step 1.30

Cancel the common factor.

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

Step 1.31

Rewrite the expression.

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

Step 1.32

Simplify.

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Step 1.32.1

Apply the distributive property.

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

Step 1.32.2

Simplify the numerator.

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Step 1.32.2.1

Simplify each term.

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Step 1.32.2.1.1

Multiply $3$ by $2$ .

$\frac{6 x + 2 \cdot - 1 - 3 x}{{(3 x - 1)}^{\frac{3}{2}}}$

Step 1.32.2.1.2

Multiply $2$ by $- 1$ .

$\frac{6 x - 2 - 3 x}{{(3 x - 1)}^{\frac{3}{2}}}$

Step 1.32.2.2

Subtract $3 x$ from $6 x$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1

Cancel the common factor.

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

Step 2.2.1.2.2

Rewrite the expression.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $3$ by $1$ .

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

Step 2.2.6

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

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

Step 2.2.7

Simplify the expression.

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Step 2.2.7.1

Add $3$ and $0$ .

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

Step 2.2.7.2

Move $3$ to the left of ${(3 x - 1)}^{\frac{3}{2}}$ .

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = 3 x - 1$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $3 x - 1$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u$ with $3 x - 1$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

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

Step 2.7.2

Subtract $2$ from $3$ .

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

Step 2.8

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

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

Step 2.9

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Multiply $3$ by $1$ .

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

Step 2.13

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 2.14

Combine fractions.

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Step 2.14.1

Add $3$ and $0$ .

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

Step 2.14.2

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

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

Step 2.14.3

Multiply $3$ by $3$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} - (3 x - 2) \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15

Simplify.

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Step 2.15.1

Apply the distributive property.

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + (- (3 x) - - 2) \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2

Simplify the numerator.

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Step 2.15.2.1

Simplify each term.

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Step 2.15.2.1.1

Multiply $3$ by $- 1$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + (- 3 x - - 2) \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.1.2

Multiply $- 1$ by $- 2$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + (- 3 x + 2) \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.2

Apply the distributive property.

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} - 3 x \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2} + 2 \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.3

Multiply $- 3 x \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}$ .

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Step 2.15.2.3.1

Combine $- 3$ and $\frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + x \frac{- 3 (9 {(3 x - 1)}^{\frac{1}{2}})}{2} + 2 \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.3.2

Multiply $9$ by $- 3$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + x \frac{- 27 {(3 x - 1)}^{\frac{1}{2}}}{2} + 2 \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.3.3

Combine $x$ and $\frac{- 27 {(3 x - 1)}^{\frac{1}{2}}}{2}$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{x (- 27 {(3 x - 1)}^{\frac{1}{2}})}{2} + 2 \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.4

Cancel the common factor of $2$ .

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Step 2.15.2.4.1

Cancel the common factor.

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{x (- 27 {(3 x - 1)}^{\frac{1}{2}})}{2} + 2 \frac{9 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.4.2

Rewrite the expression.

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{x (- 27 {(3 x - 1)}^{\frac{1}{2}})}{2} + 9 {(3 x - 1)}^{\frac{1}{2}}}{{(3 x - 1)}^{3}}$

Step 2.15.2.5

Simplify each term.

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Step 2.15.2.5.1

Simplify the numerator.

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Step 2.15.2.5.1.1

Rewrite.

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{x (- 27 {(3 x - 1)}^{\frac{1}{2}})}{2} + 9 {(3 x - 1)}^{\frac{1}{2}}}{{(3 x - 1)}^{3}}$

Step 2.15.2.5.1.2

Remove unnecessary parentheses.

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{x \cdot - 27 {(3 x - 1)}^{\frac{1}{2}}}{2} + 9 {(3 x - 1)}^{\frac{1}{2}}}{{(3 x - 1)}^{3}}$

Step 2.15.2.5.2

Move $- 27$ to the left of $x$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{- 27 \cdot x {(3 x - 1)}^{\frac{1}{2}}}{2} + 9 {(3 x - 1)}^{\frac{1}{2}}}{{(3 x - 1)}^{3}}$

Step 2.15.2.5.3

Move the negative in front of the fraction.

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} - \frac{27 x {(3 x - 1)}^{\frac{1}{2}}}{2} + 9 {(3 x - 1)}^{\frac{1}{2}}}{{(3 x - 1)}^{3}}$

Step 2.15.2.6

To write $9 {(3 x - 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} - \frac{27 x {(3 x - 1)}^{\frac{1}{2}}}{2} + 9 {(3 x - 1)}^{\frac{1}{2}} \cdot \frac{2}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.7

Combine $9 {(3 x - 1)}^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} - \frac{27 x {(3 x - 1)}^{\frac{1}{2}}}{2} + \frac{9 {(3 x - 1)}^{\frac{1}{2}} \cdot 2}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.8

Combine the numerators over the common denominator.

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{- 27 x {(3 x - 1)}^{\frac{1}{2}} + 9 {(3 x - 1)}^{\frac{1}{2}} \cdot 2}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.9

Factor $9 {(3 x - 1)}^{\frac{1}{2}}$ out of $- 27 x {(3 x - 1)}^{\frac{1}{2}} + 9 {(3 x - 1)}^{\frac{1}{2}} \cdot 2$ .

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Step 2.15.2.9.1

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

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{- 27 x {(3 x - 1)}^{\frac{1}{2}} + 9 \cdot 2 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.9.2

Factor $9 {(3 x - 1)}^{\frac{1}{2}}$ out of $- 27 x {(3 x - 1)}^{\frac{1}{2}}$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x) + 9 \cdot 2 {(3 x - 1)}^{\frac{1}{2}}}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.9.3

Factor $9 {(3 x - 1)}^{\frac{1}{2}}$ out of $9 \cdot 2 {(3 x - 1)}^{\frac{1}{2}}$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x) + 9 {(3 x - 1)}^{\frac{1}{2}} (2)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.9.4

Factor $9 {(3 x - 1)}^{\frac{1}{2}}$ out of $9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x) + 9 {(3 x - 1)}^{\frac{1}{2}} (2)$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} + \frac{9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 2)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.10

To write $3 {(3 x - 1)}^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 {(3 x - 1)}^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 2)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.11

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

$\frac{\frac{3 {(3 x - 1)}^{\frac{3}{2}} \cdot 2}{2} + \frac{9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 2)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.12

Combine the numerators over the common denominator.

$\frac{\frac{3 {(3 x - 1)}^{\frac{3}{2}} \cdot 2 + 9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 2)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.13

Rewrite $\frac{3 {(3 x - 1)}^{\frac{3}{2}} \cdot 2 + 9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 2)}{2}$ in a factored form.

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Step 2.15.2.13.1

Factor $3 {(3 x - 1)}^{\frac{1}{2}}$ out of $3 {(3 x - 1)}^{\frac{3}{2}} \cdot 2 + 9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 2)$ .

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Step 2.15.2.13.1.1

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

$\frac{\frac{3 \cdot 2 {(3 x - 1)}^{\frac{3}{2}} + 9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 2)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.13.1.2

Factor $3 {(3 x - 1)}^{\frac{1}{2}}$ out of $3 \cdot 2 {(3 x - 1)}^{\frac{3}{2}}$ .

$\frac{\frac{3 {(3 x - 1)}^{\frac{1}{2}} (2 {(3 x - 1)}^{\frac{2}{2}}) + 9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 2)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.13.1.3

Factor $3 {(3 x - 1)}^{\frac{1}{2}}$ out of $9 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 2)$ .

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

Step 2.15.2.13.1.4

Factor $3 {(3 x - 1)}^{\frac{1}{2}}$ out of $3 {(3 x - 1)}^{\frac{1}{2}} (2 {(3 x - 1)}^{\frac{2}{2}}) + 3 {(3 x - 1)}^{\frac{1}{2}} (3 (- 3 x + 2))$ .

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

Step 2.15.2.13.2

Divide $2$ by $2$ .

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

Step 2.15.2.13.3

Simplify.

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

Step 2.15.2.13.4

Apply the distributive property.

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

Step 2.15.2.13.5

Multiply $3$ by $2$ .

$\frac{\frac{3 {(3 x - 1)}^{\frac{1}{2}} (6 x + 2 \cdot - 1 + 3 (- 3 x + 2))}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.13.6

Multiply $2$ by $- 1$ .

$\frac{\frac{3 {(3 x - 1)}^{\frac{1}{2}} (6 x - 2 + 3 (- 3 x + 2))}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.13.7

Apply the distributive property.

$\frac{\frac{3 {(3 x - 1)}^{\frac{1}{2}} (6 x - 2 + 3 (- 3 x) + 3 \cdot 2)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.13.8

Multiply $- 3$ by $3$ .

$\frac{\frac{3 {(3 x - 1)}^{\frac{1}{2}} (6 x - 2 - 9 x + 3 \cdot 2)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.13.9

Multiply $3$ by $2$ .

$\frac{\frac{3 {(3 x - 1)}^{\frac{1}{2}} (6 x - 2 - 9 x + 6)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.13.10

Subtract $9 x$ from $6 x$ .

$\frac{\frac{3 {(3 x - 1)}^{\frac{1}{2}} (- 3 x - 2 + 6)}{2}}{{(3 x - 1)}^{3}}$

Step 2.15.2.13.11

Add $- 2$ and $6$ .

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

Step 2.15.3

Combine terms.

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Step 2.15.3.1

Rewrite $\frac{\frac{3 {(3 x - 1)}^{\frac{1}{2}} (- 3 x + 4)}{2}}{{(3 x - 1)}^{3}}$ as a product.

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

Step 2.15.3.2

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

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

Step 2.15.3.3

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

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

Step 2.15.3.4

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

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Step 2.15.3.4.1

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

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

Step 2.15.3.4.2

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

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

Step 2.15.3.4.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.15.3.4.4

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

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

Step 2.15.3.4.5

Combine the numerators over the common denominator.

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

Step 2.15.3.4.6

Simplify the numerator.

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Step 2.15.3.4.6.1

Multiply $3$ by $2$ .

$\frac{3 (- 3 x + 4)}{2 {(3 x - 1)}^{\frac{- 1 + 6}{2}}}$

Step 2.15.3.4.6.2

Add $- 1$ and $6$ .

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

Step 2.15.4

Factor $- 1$ out of $- 3 x$ .

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

Step 2.15.5

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

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

Step 2.15.6

Factor $- 1$ out of $- (3 x) - 1 (- 4)$ .

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

Step 2.15.7

Rewrite $- (3 x - 4)$ as $- 1 (3 x - 4)$ .

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

Step 2.15.8

Move the negative in front of the fraction.

$f'' (x) = - \frac{3 (3 x - 4)}{2 {(3 x - 1)}^{\frac{5}{2}}}$