Calculus Examples

$y = 2 x \arccos (x) - 2 \sqrt{1 - x^{2}}$

Step 1

By the Sum Rule, the derivative of $2 x \arccos (x) - 2 \sqrt{1 - x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x \arccos (x)] + \frac{d}{d x} [- 2 \sqrt{1 - x^{2}}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [2 x \arccos (x)]$ .

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Step 2.1

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

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

Step 2.2

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 $g (x) = \arccos (x)$ .

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

Step 2.3

The derivative of $\arccos (x)$ with respect to $x$ is $- \frac{1}{\sqrt{1 - x^{2}}}$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $\arccos (x)$ by $1$ .

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

Step 3

Evaluate $\frac{d}{d x} [- 2 \sqrt{1 - x^{2}}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

To apply the Chain Rule, set $u$ as $1 - x^{2}$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u$ with $1 - x^{2}$ .

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

Step 3.4

By the Sum Rule, the derivative of $1 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Combine the numerators over the common denominator.

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

Step 3.11

Simplify the numerator.

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Step 3.11.1

Multiply $- 1$ by $2$ .

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

Step 3.11.2

Subtract $2$ from $1$ .

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

Step 3.12

Move the negative in front of the fraction.

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

Step 3.13

Multiply $2$ by $- 1$ .

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

Step 3.14

Subtract $2 x$ from $0$ .

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

Step 3.15

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

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

Step 3.16

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

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

Step 3.17

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

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

Step 3.18

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

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

Step 3.19

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

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

Step 3.20

Cancel the common factors.

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Step 3.20.1

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

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

Step 3.20.2

Cancel the common factor.

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

Step 3.20.3

Rewrite the expression.

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

Step 3.21

Move the negative in front of the fraction.

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

Step 3.22

Multiply $- 1$ by $- 2$ .

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

Step 3.23

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $- 1$ by $2$ .

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

Step 4.2.2

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

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

Step 4.2.3

Move the negative in front of the fraction.

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

Step 4.3

Reorder terms.

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

Step 4.4

Simplify each term.

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Step 4.4.1

Simplify the denominator.

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Step 4.4.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 4.4.2

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

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

Step 4.4.3

Combine and simplify the denominator.

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Step 4.4.3.1

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

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

Step 4.4.3.2

Raise $\sqrt{(1 + x) (1 - x)}$ to the power of $1$ .

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

Step 4.4.3.3

Raise $\sqrt{(1 + x) (1 - x)}$ to the power of $1$ .

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

Step 4.4.3.4

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

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

Step 4.4.3.5

Add $1$ and $1$ .

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

Step 4.4.3.6

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

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Step 4.4.3.6.1

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

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

Step 4.4.3.6.2

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

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

Step 4.4.3.6.3

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

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

Step 4.4.3.6.4

Cancel the common factor of $2$ .

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Step 4.4.3.6.4.1

Cancel the common factor.

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

Step 4.4.3.6.4.2

Rewrite the expression.

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

Step 4.4.3.6.5

Simplify.

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

Write each expression with a common denominator of $(1 + x) (1 - x) {(1 - x^{2})}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

Reorder the factors of ${(1 - x^{2})}^{\frac{1}{2}} ((1 + x) (1 - x))$ .

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

Step 4.7.4

Reorder the factors of $(1 + x) (1 - x) {(1 - x^{2})}^{\frac{1}{2}}$ .

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

Step 4.8

Combine the numerators over the common denominator.

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

Step 4.9

Simplify the numerator.

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Step 4.9.1

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

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Step 4.9.1.1

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

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

Step 4.9.1.2

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

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

Step 4.9.2

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 4.9.2.1

Apply the distributive property.

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

Step 4.9.2.2

Apply the distributive property.

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

Step 4.9.2.3

Apply the distributive property.

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

Step 4.9.3

Simplify and combine like terms.

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Step 4.9.3.1

Simplify each term.

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Step 4.9.3.1.1

Multiply $1$ by $1$ .

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

Step 4.9.3.1.2

Multiply $- x$ by $1$ .

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

Step 4.9.3.1.3

Multiply $x$ by $1$ .

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

Step 4.9.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.9.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.9.3.1.5.1

Move $x$ .

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

Step 4.9.3.1.5.2

Multiply $x$ by $x$ .

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

Step 4.9.3.2

Add $- x$ and $x$ .

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

Step 4.9.3.3

Add $1$ and $0$ .

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

Step 4.10

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

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

Step 4.11

Write each expression with a common denominator of $(1 + x) (1 - x) {(1 - x^{2})}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.11.1

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

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

Step 4.11.2

Reorder the factors of $(1 + x) (1 - x) {(1 - x^{2})}^{\frac{1}{2}}$ .

Step 4.12

Combine the numerators over the common denominator.

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

Step 4.13

Simplify the numerator.

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Step 4.13.1

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

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Step 4.13.1.1

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

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

Step 4.13.1.2

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

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

Step 4.13.1.3

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

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

Step 4.13.2

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

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

Step 4.13.3

Simplify each term.

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Step 4.13.3.1

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 4.13.3.1.1

Apply the distributive property.

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

Step 4.13.3.1.2

Apply the distributive property.

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

Step 4.13.3.1.3

Apply the distributive property.

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

Step 4.13.3.2

Simplify and combine like terms.

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Step 4.13.3.2.1

Simplify each term.

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Step 4.13.3.2.1.1

Multiply $1$ by $1$ .

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

Step 4.13.3.2.1.2

Multiply $- x$ by $1$ .

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

Step 4.13.3.2.1.3

Multiply $x$ by $1$ .

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

Step 4.13.3.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.13.3.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.13.3.2.1.5.1

Move $x$ .

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

Step 4.13.3.2.1.5.2

Multiply $x$ by $x$ .

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

Step 4.13.3.2.2

Add $- x$ and $x$ .

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

Step 4.13.3.2.3

Add $1$ and $0$ .

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

Step 4.13.3.3

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

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Step 4.13.3.3.1

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

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

Step 4.13.3.3.2

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

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

Step 4.13.3.3.3

Combine the numerators over the common denominator.

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

Step 4.13.3.3.4

Add $1$ and $1$ .

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

Step 4.13.3.3.5

Divide $2$ by $2$ .

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

Step 4.13.3.4

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

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

Step 4.13.3.5

Apply the distributive property.

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

Step 4.13.3.6

Multiply $- 1$ by $1$ .

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

Step 4.13.3.7

Multiply $- - x^{2}$ .

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Step 4.13.3.7.1

Multiply $- 1$ by $- 1$ .

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

Step 4.13.3.7.2

Multiply $x^{2}$ by $1$ .

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

Step 4.13.4

Combine the opposite terms in $1 - x^{2} - 1 + x^{2}$ .

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Step 4.13.4.1

Subtract $1$ from $1$ .

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

Step 4.13.4.2

Add $- x^{2}$ and $0$ .

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

Step 4.13.4.3

Add $- x^{2}$ and $x^{2}$ .

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

Step 4.13.5

Multiply $x$ by $0$ .

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

Step 4.13.6

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 4.13.6.1

Apply the distributive property.

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

Step 4.13.6.2

Apply the distributive property.

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

Step 4.13.6.3

Apply the distributive property.

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

Step 4.13.7

Simplify and combine like terms.

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Step 4.13.7.1

Simplify each term.

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Step 4.13.7.1.1

Multiply $1$ by $1$ .

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

Step 4.13.7.1.2

Multiply $- x$ by $1$ .

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

Step 4.13.7.1.3

Multiply $x$ by $1$ .

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

Step 4.13.7.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.13.7.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.13.7.1.5.1

Move $x$ .

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

Step 4.13.7.1.5.2

Multiply $x$ by $x$ .

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

Step 4.13.7.2

Add $- x$ and $x$ .

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

Step 4.13.7.3

Add $1$ and $0$ .

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

Step 4.13.8

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

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Step 4.13.8.1

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

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Step 4.13.8.1.1

Raise $1 - x^{2}$ to the power of $1$ .

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

Step 4.13.8.1.2

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

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

Step 4.13.8.2

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

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

Step 4.13.8.3

Combine the numerators over the common denominator.

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

Step 4.13.8.4

Add $2$ and $1$ .

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

Step 4.13.9

Add $0$ and $\arccos (x) {(1 - x^{2})}^{\frac{3}{2}}$ .

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

Step 4.14

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

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

Step 4.15

Simplify the numerator.

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Step 4.15.1

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

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Step 4.15.1.1

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

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

Step 4.15.1.2

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

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

Step 4.15.1.3

Combine the numerators over the common denominator.

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

Step 4.15.1.4

Add $- 1$ and $3$ .

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

Step 4.15.1.5

Divide $2$ by $2$ .

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

Step 4.15.2

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

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

Step 4.16

Simplify the numerator.

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Step 4.16.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.16.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 4.17

Cancel the common factor of $1 + x$ .

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Step 4.17.1

Cancel the common factor.

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

Step 4.17.2

Rewrite the expression.

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

Step 4.18

Cancel the common factor of $1 - x$ .

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Step 4.18.1

Cancel the common factor.

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

Step 4.18.2

Divide $2 \arccos (x)$ by $1$ .

$2 \arccos (x)$