Calculus Examples

$x \sqrt{x - 1}$

Step 1

Find the first derivative.

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Step 1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x - 1}$ as

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

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

Step 1.2

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

$f (x) = x$ and

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

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

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

$g (x) = x - 1$ .

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Step 1.3.1

To apply the Chain Rule, set

$u$ as

$x - 1$ .

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

Step 1.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = \frac{1}{2}$ .

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

Step 1.3.3

Replace all occurrences of

$u$ with

$x - 1$ .

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

Step 1.4

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 1.5

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply

$- 1$ by

$2$ .

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

Step 1.7.2

Subtract

$2$ from

$1$ .

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

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

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

Step 1.8.2

Combine

$\frac{1}{2}$ and

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

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

Step 1.8.3

Move

${(x - 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.8.4

Combine

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

$x$ .

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

Step 1.9

By the Sum Rule, the derivative of

$x - 1$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.10

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.11

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- 1$ with respect to

$x$ is

$0$ .

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

Step 1.12

Simplify the expression.

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Step 1.12.1

Add

$1$ and

$0$ .

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

Step 1.12.2

Multiply

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

$1$ .

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

Step 1.13

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 1.14

Multiply

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

$1$ .

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

Step 1.15

To write

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

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

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

Step 1.16

Combine

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

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

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

Step 1.17

Combine the numerators over the common denominator.

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

Step 1.18

Multiply

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

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

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Step 1.18.1

Move

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

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

Step 1.18.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.18.3

Combine the numerators over the common denominator.

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

Step 1.18.4

Add

$1$ and

$1$ .

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

Step 1.18.5

Divide

$2$ by

$2$ .

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

Step 1.19

Simplify

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

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

Step 1.20

Move

$2$ to the left of

$x - 1$ .

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

Step 1.21

Simplify.

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Step 1.21.1

Apply the distributive property.

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

Step 1.21.2

Simplify the numerator.

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Step 1.21.2.1

Multiply

$2$ by

$- 1$ .

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

Step 1.21.2.2

Add

$x$ and

$2 x$ .

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

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

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

$x$ is

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

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

Step 2.2

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

$f (x) = 3 x - 2$ and

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

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

Step 2.3

Multiply the exponents in

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

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Step 2.3.1

Apply the power rule and multiply exponents,

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

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

Step 2.3.2

Cancel the common factor of

$2$ .

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Step 2.3.2.1

Cancel the common factor.

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

Step 2.3.2.2

Rewrite the expression.

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

Step 2.4

Simplify.

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

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of

$3 x - 2$ with respect to

$x$ is

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]$ .

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

Step 2.5.2

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

$3 \frac{d}{d x} [x]$ .

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

Step 2.5.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 2.5.4

Multiply

$3$ by

$1$ .

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

Step 2.5.5

Since

$- 2$ is constant with respect to

$x$ , the derivative of

$- 2$ with respect to

$x$ is

$0$ .

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

Step 2.5.6

Simplify the expression.

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Step 2.5.6.1

Add

$3$ and

$0$ .

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

Step 2.5.6.2

Move

$3$ to the left of

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

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

Step 2.6

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

$g (x) = x - 1$ .

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Step 2.6.1

To apply the Chain Rule, set

$u_{1}$ as

$x - 1$ .

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

Step 2.6.2

Differentiate using the Power Rule which states that

$\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is

$n {u_{1}}^{n - 1}$ where

$n = \frac{1}{2}$ .

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

Step 2.6.3

Replace all occurrences of

$u_{1}$ with

$x - 1$ .

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

Step 2.7

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 2.8

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply

$- 1$ by

$2$ .

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

Step 2.10.2

Subtract

$2$ from

$1$ .

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

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

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

Step 2.11.2

Combine

$\frac{1}{2}$ and

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

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

Step 2.11.3

Move

${(x - 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.12

By the Sum Rule, the derivative of

$x - 1$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 2.13

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 2.14

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- 1$ with respect to

$x$ is

$0$ .

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

Step 2.15

Combine fractions.

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Step 2.15.1

Add

$1$ and

$0$ .

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

Step 2.15.2

Multiply

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

$1$ .

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

Step 2.15.3

Multiply

$\frac{1}{2}$ by

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

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

Step 2.16

Simplify.

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Step 2.16.1

Apply the distributive property.

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

Step 2.16.2

Apply the distributive property.

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

Step 2.16.3

Simplify the numerator.

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Step 2.16.3.1

Add parentheses.

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

Step 2.16.3.2

Let

$u_{2} = {(x - 1)}^{\frac{1}{2}}$ . Substitute

$u_{2}$ for all occurrences of

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

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Step 2.16.3.2.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{3 \cdot 2 u_{2} \cdot u_{2} - 3 x + 2}{2 u_{2}}}{2 x + 2 \cdot - 1}$

Step 2.16.3.2.2

Multiply

$u_{2}$ by

$u_{2}$ by adding the exponents.

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Step 2.16.3.2.2.1

Move

$u_{2}$ .

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

Step 2.16.3.2.2.2

Multiply

$u_{2}$ by

$u_{2}$ .

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

Step 2.16.3.2.3

Multiply

$3$ by

$2$ .

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

Step 2.16.3.3

Replace all occurrences of

$u_{2}$ with

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

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

Step 2.16.3.4

Simplify.

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Step 2.16.3.4.1

Simplify each term.

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Step 2.16.3.4.1.1

Multiply the exponents in

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

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Step 2.16.3.4.1.1.1

Apply the power rule and multiply exponents,

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

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

Step 2.16.3.4.1.1.2

Cancel the common factor of

$2$ .

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Step 2.16.3.4.1.1.2.1

Cancel the common factor.

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

Step 2.16.3.4.1.1.2.2

Rewrite the expression.

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

Step 2.16.3.4.1.2

Simplify.

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

Step 2.16.3.4.1.3

Apply the distributive property.

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

Step 2.16.3.4.1.4

Multiply

$6$ by

$- 1$ .

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

Step 2.16.3.4.2

Subtract

$3 x$ from

$6 x$ .

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

Step 2.16.3.4.3

Add

$- 6$ and

$2$ .

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

Step 2.16.4

Combine terms.

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Step 2.16.4.1

Multiply

$2$ by

$- 1$ .

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

Step 2.16.4.2

Rewrite

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

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

Step 2.16.4.3

Multiply

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

$\frac{1}{2 x - 2}$ .

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

Step 2.16.5

Simplify the denominator.

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Step 2.16.5.1

Factor

$2$ out of

$2 x - 2$ .

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Step 2.16.5.1.1

Factor

$2$ out of

$2 x$ .

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

Step 2.16.5.1.2

Factor

$2$ out of

$- 2$ .

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

Step 2.16.5.1.3

Factor

$2$ out of

$2 (x) + 2 (- 1)$ .

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

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

Step 2.16.5.2

Combine exponents.

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Step 2.16.5.2.1

Multiply

$2$ by

$2$ .

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

Step 2.16.5.2.2

Raise

$x - 1$ to the power of

$1$ .

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

Step 2.16.5.2.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.16.5.2.4

Write

$1$ as a fraction with a common denominator.

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

Step 2.16.5.2.5

Combine the numerators over the common denominator.

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

Step 2.16.5.2.6

Add

$2$ and

$1$ .

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

Step 3

Find the third derivative.

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Step 3.1

Since

$\frac{1}{4}$ is constant with respect to

$x$ , the derivative of

$\frac{3 x - 4}{4 {(x - 1)}^{\frac{3}{2}}}$ with respect to

$x$ is

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

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

Step 3.2

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

$f (x) = 3 x - 4$ and

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

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

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in

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

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Step 3.3.1.1

Apply the power rule and multiply exponents,

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

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

Step 3.3.1.2

Cancel the common factor of

$2$ .

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Step 3.3.1.2.1

Cancel the common factor.

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

Step 3.3.1.2.2

Rewrite the expression.

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

Step 3.3.2

By the Sum Rule, the derivative of

$3 x - 4$ with respect to

$x$ is

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 4]$ .

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

Step 3.3.3

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

$3 \frac{d}{d x} [x]$ .

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

Step 3.3.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 3.3.5

Multiply

$3$ by

$1$ .

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

Step 3.3.6

Since

$- 4$ is constant with respect to

$x$ , the derivative of

$- 4$ with respect to

$x$ is

$0$ .

$\frac{1}{4} \cdot \frac{{(x - 1)}^{\frac{3}{2}} (3 + 0) - (3 x - 4) \frac{d}{d x} [{(x - 1)}^{\frac{3}{2}}]}{{(x - 1)}^{3}}$

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Add

$3$ and

$0$ .

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

Step 3.3.7.2

Move

$3$ to the left of

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

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

Step 3.4

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

$g (x) = x - 1$ .

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Step 3.4.1

To apply the Chain Rule, set

$u$ as

$x - 1$ .

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

Step 3.4.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = \frac{3}{2}$ .

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

Step 3.4.3

Replace all occurrences of

$u$ with

$x - 1$ .

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

Step 3.5

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 3.6

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply

$- 1$ by

$2$ .

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

Step 3.8.2

Subtract

$2$ from

$3$ .

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

Step 3.9

Combine

$\frac{3}{2}$ and

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

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

Step 3.10

By the Sum Rule, the derivative of

$x - 1$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 3.11

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 3.12

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- 1$ with respect to

$x$ is

$0$ .

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

Step 3.13

Combine fractions.

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Step 3.13.1

Add

$1$ and

$0$ .

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

Step 3.13.2

Multiply

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

$1$ .

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

Step 3.13.3

Multiply

$\frac{1}{4}$ by

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

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

Step 3.14

Simplify.

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Step 3.14.1

Apply the distributive property.

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

Step 3.14.2

Simplify the numerator.

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Step 3.14.2.1

Simplify each term.

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Step 3.14.2.1.1

Multiply

$3$ by

$- 1$ .

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

Step 3.14.2.1.2

Multiply

$- 1$ by

$- 4$ .

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

Step 3.14.2.2

Apply the distributive property.

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

Step 3.14.2.3

Multiply

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

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Step 3.14.2.3.1

Combine

$- 3$ and

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

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

Step 3.14.2.3.2

Multiply

$3$ by

$- 3$ .

$\frac{3 {(x - 1)}^{\frac{3}{2}} + x \frac{- 9 {(x - 1)}^{\frac{1}{2}}}{2} + 4 \frac{3 {(x - 1)}^{\frac{1}{2}}}{2}}{4 {(x - 1)}^{3}}$

Step 3.14.2.3.3

Combine

$x$ and

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

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{x (- 9 {(x - 1)}^{\frac{1}{2}})}{2} + 4 \frac{3 {(x - 1)}^{\frac{1}{2}}}{2}}{4 {(x - 1)}^{3}}$

Step 3.14.2.4

Cancel the common factor of

$2$ .

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Step 3.14.2.4.1

Factor

$2$ out of

$4$ .

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{x (- 9 {(x - 1)}^{\frac{1}{2}})}{2} + 2 (2) \frac{3 {(x - 1)}^{\frac{1}{2}}}{2}}{4 {(x - 1)}^{3}}$

Step 3.14.2.4.2

Cancel the common factor.

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{x (- 9 {(x - 1)}^{\frac{1}{2}})}{2} + 2 \cdot 2 \frac{3 {(x - 1)}^{\frac{1}{2}}}{2}}{4 {(x - 1)}^{3}}$

Step 3.14.2.4.3

Rewrite the expression.

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{x (- 9 {(x - 1)}^{\frac{1}{2}})}{2} + 2 (3 {(x - 1)}^{\frac{1}{2}})}{4 {(x - 1)}^{3}}$

Step 3.14.2.5

Multiply

$3$ by

$2$ .

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{x (- 9 {(x - 1)}^{\frac{1}{2}})}{2} + 6 {(x - 1)}^{\frac{1}{2}}}{4 {(x - 1)}^{3}}$

Step 3.14.2.6

Simplify each term.

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Step 3.14.2.6.1

Simplify the numerator.

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Step 3.14.2.6.1.1

Rewrite.

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{x (- 9 {(x - 1)}^{\frac{1}{2}})}{2} + 6 {(x - 1)}^{\frac{1}{2}}}{4 {(x - 1)}^{3}}$

Step 3.14.2.6.1.2

Remove unnecessary parentheses.

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{x \cdot - 9 {(x - 1)}^{\frac{1}{2}}}{2} + 6 {(x - 1)}^{\frac{1}{2}}}{4 {(x - 1)}^{3}}$

Step 3.14.2.6.2

Move

$- 9$ to the left of

$x$ .

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{- 9 \cdot x {(x - 1)}^{\frac{1}{2}}}{2} + 6 {(x - 1)}^{\frac{1}{2}}}{4 {(x - 1)}^{3}}$

Step 3.14.2.6.3

Move the negative in front of the fraction.

$\frac{3 {(x - 1)}^{\frac{3}{2}} - \frac{9 x {(x - 1)}^{\frac{1}{2}}}{2} + 6 {(x - 1)}^{\frac{1}{2}}}{4 {(x - 1)}^{3}}$

Step 3.14.2.7

To write

$6 {(x - 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\frac{3 {(x - 1)}^{\frac{3}{2}} - \frac{9 x {(x - 1)}^{\frac{1}{2}}}{2} + 6 {(x - 1)}^{\frac{1}{2}} \cdot \frac{2}{2}}{4 {(x - 1)}^{3}}$

Step 3.14.2.8

Combine

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

$\frac{2}{2}$ .

$\frac{3 {(x - 1)}^{\frac{3}{2}} - \frac{9 x {(x - 1)}^{\frac{1}{2}}}{2} + \frac{6 {(x - 1)}^{\frac{1}{2}} \cdot 2}{2}}{4 {(x - 1)}^{3}}$

Step 3.14.2.9

Combine the numerators over the common denominator.

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{- 9 x {(x - 1)}^{\frac{1}{2}} + 6 {(x - 1)}^{\frac{1}{2}} \cdot 2}{2}}{4 {(x - 1)}^{3}}$

Step 3.14.2.10

Simplify the numerator.

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Step 3.14.2.10.1

Factor

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

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

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Step 3.14.2.10.1.1

Move

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

$\frac{3 {(x - 1)}^{\frac{3}{2}} + \frac{- 9 x {(x - 1)}^{\frac{1}{2}} + 6 \cdot 2 {(x - 1)}^{\frac{1}{2}}}{2}}{4 {(x - 1)}^{3}}$

Step 3.14.2.10.1.2

Factor

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

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

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

Step 3.14.2.10.1.3

Factor

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

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

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

Step 3.14.2.10.1.4

Factor

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

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

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

Step 3.14.2.10.2

Multiply

$2$ by

$2$ .

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

Step 3.14.2.11

To write

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

$\frac{2}{2}$ .

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

Step 3.14.2.12

Combine

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

$\frac{2}{2}$ .

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

Step 3.14.2.13

Combine the numerators over the common denominator.

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

Step 3.14.2.14

Rewrite

$\frac{3 {(x - 1)}^{\frac{3}{2}} \cdot 2 + 3 {(x - 1)}^{\frac{1}{2}} (- 3 x + 4)}{2}$ in a factored form.

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Step 3.14.2.14.1

Factor

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

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

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Step 3.14.2.14.1.1

Move

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

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

Step 3.14.2.14.1.2

Factor

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

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

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

Step 3.14.2.14.1.3

Factor

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

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

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

Step 3.14.2.14.2

Divide

$2$ by

$2$ .

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

Step 3.14.2.14.3

Simplify.

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

Step 3.14.2.14.4

Apply the distributive property.

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

Step 3.14.2.14.5

Multiply

$2$ by

$- 1$ .

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

Step 3.14.2.14.6

Subtract

$3 x$ from

$2 x$ .

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

Step 3.14.2.14.7

Add

$- 2$ and

$4$ .

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

Step 3.14.3

Combine terms.

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Step 3.14.3.1

Rewrite

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

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

Step 3.14.3.2

Multiply

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

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

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

Step 3.14.3.3

Multiply

$4$ by

$2$ .

$\frac{3 {(x - 1)}^{\frac{1}{2}} (- x + 2)}{8 {(x - 1)}^{3}}$

Step 3.14.3.4

Move

${(x - 1)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{n} = \frac{1}{b^{- n}}$ .

$\frac{3 (- x + 2)}{8 {(x - 1)}^{3} {(x - 1)}^{- \frac{1}{2}}}$

Step 3.14.3.5

Multiply

${(x - 1)}^{3}$ by

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

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Step 3.14.3.5.1

Move

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

$\frac{3 (- x + 2)}{8 ({(x - 1)}^{- \frac{1}{2}} {(x - 1)}^{3})}$

Step 3.14.3.5.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (- x + 2)}{8 {(x - 1)}^{- \frac{1}{2} + 3}}$

Step 3.14.3.5.3

To write

$3$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 3.14.3.5.4

Combine

$3$ and

$\frac{2}{2}$ .

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

Step 3.14.3.5.5

Combine the numerators over the common denominator.

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

Step 3.14.3.5.6

Simplify the numerator.

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Step 3.14.3.5.6.1

Multiply

$3$ by

$2$ .

$\frac{3 (- x + 2)}{8 {(x - 1)}^{\frac{- 1 + 6}{2}}}$

Step 3.14.3.5.6.2

Add

$- 1$ and

$6$ .

$\frac{3 (- x + 2)}{8 {(x - 1)}^{\frac{5}{2}}}$

Step 3.14.4

Factor

$- 1$ out of

$- x$ .

$\frac{3 (- (x) + 2)}{8 {(x - 1)}^{\frac{5}{2}}}$

Step 3.14.5

Rewrite

$2$ as

$- 1 (- 2)$ .

$\frac{3 (- (x) - 1 (- 2))}{8 {(x - 1)}^{\frac{5}{2}}}$

Step 3.14.6

Factor

$- 1$ out of

$- (x) - 1 (- 2)$ .

$\frac{3 (- (x - 2))}{8 {(x - 1)}^{\frac{5}{2}}}$

Step 3.14.7

Rewrite

$- (x - 2)$ as

$- 1 (x - 2)$ .

$\frac{3 (- 1 (x - 2))}{8 {(x - 1)}^{\frac{5}{2}}}$

Step 3.14.8

Move the negative in front of the fraction.

$f''' (x) = - \frac{3 (x - 2)}{8 {(x - 1)}^{\frac{5}{2}}}$