Calculus Examples

$f (x) = \ln (\sqrt{\frac{{(x - 1)}^{3}}{x + 1}})$

Step 1

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as ${(\frac{{(x - 1)}^{3}}{x + 1})}^{\frac{1}{2}}$ .

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

Step 2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 2.3

Replace all occurrences of $u_{1}$ with ${(\frac{{(x - 1)}^{3}}{x + 1})}^{\frac{1}{2}}$ .

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

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

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $\frac{{(x - 1)}^{3}}{x + 1}$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u_{2}$ with $\frac{{(x - 1)}^{3}}{x + 1}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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 $f (x) = {(x - 1)}^{3}$ and $g (x) = x + 1$ .

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

Step 10

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 $g (x) = x - 1$ .

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Step 10.1

To apply the Chain Rule, set $u_{3}$ as $x - 1$ .

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

Step 10.2

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

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

Step 10.3

Replace all occurrences of $u_{3}$ with $x - 1$ .

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

Step 11

Differentiate.

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Step 11.1

Move $3$ to the left of $x + 1$ .

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

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

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

Step 11.5

Simplify the expression.

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Step 11.5.1

Add $1$ and $0$ .

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

Step 11.5.2

Multiply $3$ by $1$ .

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

Step 11.6

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

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

Step 11.7

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

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

Step 11.8

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

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

Step 11.9

Combine fractions.

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Step 11.9.1

Add $1$ and $0$ .

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

Step 11.9.2

Multiply $- 1$ by $1$ .

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

Step 11.9.3

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

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

Step 11.9.4

Move $2$ to the left of ${(x + 1)}^{2}$ .

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

Step 12

Simplify.

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Step 12.1

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 12.2

Apply the product rule to $\frac{{(x - 1)}^{3}}{x + 1}$ .

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

Step 12.3

Apply the product rule to $\frac{x + 1}{{(x - 1)}^{3}}$ .

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

Step 12.4

Apply the distributive property.

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

Step 12.5

Combine terms.

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Step 12.5.1

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

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Step 12.5.1.1

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

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

Step 12.5.1.2

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

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

Step 12.5.2

Multiply by the reciprocal of the fraction to divide by $\frac{{(x - 1)}^{\frac{3}{2}}}{{(x + 1)}^{\frac{1}{2}}}$ .

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

Step 12.5.3

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

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

Step 12.5.4

Multiply $3$ by $1$ .

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

Step 12.5.5

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

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Step 12.5.5.1

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

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

Step 12.5.5.2

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

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

Step 12.5.6

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

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

Step 12.5.7

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

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

Step 12.5.8

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

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Step 12.5.8.1

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

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

Step 12.5.8.2

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

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

Step 12.5.8.3

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

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

Step 12.5.8.4

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

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

Step 12.5.8.5

Combine the numerators over the common denominator.

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

Step 12.5.8.6

Simplify the numerator.

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Step 12.5.8.6.1

Multiply $2$ by $2$ .

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

Step 12.5.8.6.2

Add $- 1$ and $4$ .

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

Step 12.5.9

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

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

Step 12.5.10

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

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Step 12.5.10.1

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

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

Step 12.5.10.2

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

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

Step 12.5.10.3

Combine the numerators over the common denominator.

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

Step 12.5.10.4

Add $3$ and $3$ .

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

Step 12.5.10.5

Divide $6$ by $2$ .

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

Step 12.5.11

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

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

Step 12.5.12

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

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

Step 12.5.13

Simplify the denominator.

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Step 12.5.13.1

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

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Step 12.5.13.1.1

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

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

Step 12.5.13.1.2

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

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

Step 12.5.13.1.3

Combine the numerators over the common denominator.

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

Step 12.5.13.1.4

Add $- 1$ and $3$ .

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

Step 12.5.13.1.5

Divide $2$ by $2$ .

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

Step 12.5.13.2

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

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

Step 12.6

Reorder terms.

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

Step 12.7

Simplify the numerator.

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Step 12.7.1

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

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Step 12.7.1.1

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

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

Step 12.7.1.2

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

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

Step 12.7.2

Apply the distributive property.

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

Step 12.7.3

Multiply $- 1$ by $- 1$ .

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

Step 12.7.4

Add $- x$ and $3 x$ .

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

Step 12.7.5

Add $1$ and $3$ .

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

Step 12.7.6

Factor $2$ out of $2 x + 4$ .

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Step 12.7.6.1

Factor $2$ out of $2 x$ .

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

Step 12.7.6.2

Factor $2$ out of $4$ .

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

Step 12.7.6.3

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

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

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

Step 12.8

Cancel the common factor of ${(x - 1)}^{2}$ and ${(x - 1)}^{3}$ .

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Step 12.8.1

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

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

Step 12.8.2

Cancel the common factors.

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Step 12.8.2.1

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

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

Step 12.8.2.2

Cancel the common factor.

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

Step 12.8.2.3

Rewrite the expression.

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

Step 12.9

Cancel the common factor of $2$ .

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Step 12.9.1

Cancel the common factor.

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

Step 12.9.2

Rewrite the expression.

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