Calculus Examples

$y = \arccos (\sqrt{x}) + \arcsin (\sqrt{1 - x})$

Step 1

Rewrite the right side with rational exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arccos (x^{\frac{1}{2}}) + \arcsin ({(1 - x)}^{\frac{1}{2}}))$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

Evaluate $\frac{d}{d x} [\arccos (x^{\frac{1}{2}})]$ .

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Step 4.2.1

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

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Step 4.2.1.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{1}{2}}$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Replace all occurrences of $u_{1}$ with $x^{\frac{1}{2}}$ .

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

Step 4.2.2

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

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

Step 4.2.3

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

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Step 4.2.3.1

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

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

Step 4.2.3.2

Cancel the common factor of $2$ .

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Step 4.2.3.2.1

Cancel the common factor.

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

Step 4.2.3.2.2

Rewrite the expression.

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

Step 4.2.4

Simplify.

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

Step 4.2.5

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

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

Step 4.2.6

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

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

Step 4.2.7

Combine the numerators over the common denominator.

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

Step 4.2.8

Simplify the numerator.

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Step 4.2.8.1

Multiply $- 1$ by $2$ .

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

Step 4.2.8.2

Subtract $2$ from $1$ .

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

Step 4.2.9

Move the negative in front of the fraction.

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

Step 4.2.10

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

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

Step 4.2.11

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

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

Step 4.2.12

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

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

Step 4.3

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

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Step 4.3.1

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

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Step 4.3.1.1

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

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

Step 4.3.1.2

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

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

Step 4.3.1.3

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

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

Step 4.3.2

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

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Step 4.3.2.1

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

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

Step 4.3.2.2

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

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

Step 4.3.2.3

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

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

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

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

Step 4.3.7

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

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Step 4.3.7.1

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

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

Step 4.3.7.2

Cancel the common factor of $2$ .

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Step 4.3.7.2.1

Cancel the common factor.

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

Step 4.3.7.2.2

Rewrite the expression.

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

Step 4.3.8

Simplify.

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

Step 4.3.9

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

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

Step 4.3.10

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

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

Step 4.3.11

Combine the numerators over the common denominator.

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

Step 4.3.12

Simplify the numerator.

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Step 4.3.12.1

Multiply $- 1$ by $2$ .

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

Step 4.3.12.2

Subtract $2$ from $1$ .

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

Step 4.3.13

Move the negative in front of the fraction.

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

Step 4.3.14

Multiply $- 1$ by $1$ .

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

Step 4.3.15

Subtract $1$ from $0$ .

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

Step 4.3.16

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

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

Step 4.3.17

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

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

Step 4.3.18

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

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

Step 4.3.19

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

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

Step 4.3.20

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

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

Step 4.3.21

Move the negative in front of the fraction.

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

Step 4.3.22

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

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

Step 4.4

Simplify.

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Step 4.4.1

Apply the distributive property.

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

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Multiply $- 1$ by $1$ .

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

Step 4.4.2.2

Multiply $- 1$ by $- 1$ .

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

Step 4.4.2.3

Multiply $x$ by $1$ .

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

Step 4.4.2.4

Subtract $1$ from $1$ .

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

Step 4.4.2.5

Add $0$ and $x$ .

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

Step 4.4.3

Simplify each term.

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Step 4.4.3.1

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

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

Step 4.4.3.2

Combine and simplify the denominator.

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Step 4.4.3.2.1

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

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

Step 4.4.3.2.2

Move $\sqrt{1 - x}$ .

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

Step 4.4.3.2.3

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

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

Step 4.4.3.2.4

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

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

Step 4.4.3.2.5

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

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

Step 4.4.3.2.6

Add $1$ and $1$ .

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

Step 4.4.3.2.7

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

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Step 4.4.3.2.7.1

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

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

Step 4.4.3.2.7.2

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

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

Step 4.4.3.2.7.3

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

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

Step 4.4.3.2.7.4

Cancel the common factor of $2$ .

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Step 4.4.3.2.7.4.1

Cancel the common factor.

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

Step 4.4.3.2.7.4.2

Rewrite the expression.

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

Step 4.4.3.2.7.5

Simplify.

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

Step 4.4.3.3

Simplify the denominator.

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Step 4.4.3.3.1

Rewrite.

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

Step 4.4.3.3.2

Apply the distributive property.

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

Step 4.4.3.3.3

Multiply $- 1$ by $1$ .

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

Step 4.4.3.3.4

Multiply $- 1$ by $- 1$ .

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

Step 4.4.3.3.5

Multiply $x$ by $1$ .

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

Step 4.4.3.3.6

Subtract $1$ from $1$ .

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

Step 4.4.3.3.7

Add $0$ and $x$ .

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

Step 4.4.3.3.8

Remove unnecessary parentheses.

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

Step 4.4.3.4

Move $2$ to the left of $\sqrt{x}$ .

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

Step 4.4.3.5

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

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

Step 4.4.3.6

Combine and simplify the denominator.

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Step 4.4.3.6.1

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

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

Step 4.4.3.6.2

Move $\sqrt{x}$ .

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

Step 4.4.3.6.3

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 4.4.3.6.4

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 4.4.3.6.5

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

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

Step 4.4.3.6.6

Add $1$ and $1$ .

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

Step 4.4.3.6.7

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 4.4.3.6.7.1

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

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

Step 4.4.3.6.7.2

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

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

Step 4.4.3.6.7.3

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

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

Step 4.4.3.6.7.4

Cancel the common factor of $2$ .

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Step 4.4.3.6.7.4.1

Cancel the common factor.

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

Step 4.4.3.6.7.4.2

Rewrite the expression.

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

Step 4.4.3.6.7.5

Simplify.

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

Step 4.4.4

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

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

Step 4.4.5

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

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

Step 4.4.6

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

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Step 4.4.6.1

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

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

Step 4.4.6.2

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

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

Step 4.4.6.3

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

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

Step 4.4.6.4

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

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

Step 4.4.6.5

Combine the numerators over the common denominator.

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

Step 4.4.6.6

Add $2$ and $1$ .

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

Step 4.4.6.7

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

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Step 4.4.6.7.1

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

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

Step 4.4.6.7.2

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

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Step 4.4.6.7.2.1

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

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

Step 4.4.6.7.2.2

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

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

Step 4.4.6.7.3

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

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

Step 4.4.6.7.4

Combine the numerators over the common denominator.

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

Step 4.4.6.7.5

Add $1$ and $2$ .

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

Step 4.4.6.8

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

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

Step 4.4.6.9

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

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

Step 4.4.6.10

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

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

Step 4.4.6.11

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

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

Step 4.4.6.12

Combine the numerators over the common denominator.

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

Step 4.4.6.13

Add $1$ and $2$ .

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

Step 4.4.6.14

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

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

Step 4.4.6.15

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

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

Step 4.4.6.16

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

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

Step 4.4.6.17

Combine the numerators over the common denominator.

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

Step 4.4.6.18

Add $1$ and $2$ .

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

Step 4.4.7

Combine the numerators over the common denominator.

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

Step 4.4.8

Simplify the numerator.

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Step 4.4.8.1

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

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

Step 4.4.8.2

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

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

Step 4.4.8.3

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

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Step 4.4.8.3.1

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

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

Step 4.4.8.3.2

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

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

Step 4.4.8.3.3

Combine the numerators over the common denominator.

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

Step 4.4.8.3.4

Add $1$ and $1$ .

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

Step 4.4.8.3.5

Divide $2$ by $2$ .

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

Step 4.4.8.4

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

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

Step 4.4.8.5

Apply the distributive property.

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

Step 4.4.8.6

Multiply $- 1$ by $1$ .

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

Step 4.4.8.7

Multiply $- - x$ .

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Step 4.4.8.7.1

Multiply $- 1$ by $- 1$ .

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

Step 4.4.8.7.2

Multiply $x$ by $1$ .

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

Step 4.4.8.8

Apply the distributive property.

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

Step 4.4.8.9

Rewrite $- 1 x$ as $- x$ .

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

Step 4.4.8.10

Multiply $x$ by $x$ .

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

Step 4.4.8.11

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

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Step 4.4.8.11.1

Move $x^{\frac{1}{2}}$ .

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

Step 4.4.8.11.2

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

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

Step 4.4.8.11.3

Combine the numerators over the common denominator.

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

Step 4.4.8.11.4

Add $1$ and $1$ .

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

Step 4.4.8.11.5

Divide $2$ by $2$ .

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

Step 4.4.8.12

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

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

Step 4.4.8.13

Apply the distributive property.

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

Step 4.4.8.14

Multiply $- 1$ by $1$ .

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

Step 4.4.8.15

Rewrite using the commutative property of multiplication.

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

Step 4.4.8.16

Simplify each term.

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Step 4.4.8.16.1

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

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Step 4.4.8.16.1.1

Move $x$ .

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

Step 4.4.8.16.1.2

Multiply $x$ by $x$ .

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

Step 4.4.8.16.2

Multiply $- 1$ by $- 1$ .

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

Step 4.4.8.16.3

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

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

Step 4.4.8.17

Subtract $x$ from $- x$ .

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

Step 4.4.8.18

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

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

Step 4.4.8.19

Factor $2 x$ out of $2 x^{2} - 2 x$ .

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Step 4.4.8.19.1

Factor $2 x$ out of $2 x^{2}$ .

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

Step 4.4.8.19.2

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

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

Step 4.4.8.19.3

Factor $2 x$ out of $2 x (x) + 2 x (- 1)$ .

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

Step 4.4.9

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

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

Step 4.4.10

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

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Step 4.4.10.1

Move $x^{- 1}$ .

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

Step 4.4.10.2

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

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

Step 4.4.10.3

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

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

Step 4.4.10.4

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

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

Step 4.4.10.5

Combine the numerators over the common denominator.

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

Step 4.4.10.6

Simplify the numerator.

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Step 4.4.10.6.1

Multiply $- 1$ by $2$ .

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

Step 4.4.10.6.2

Add $- 2$ and $3$ .

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

Step 4.4.11

Cancel the common factor.

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

Step 4.4.12

Rewrite the expression.

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

Step 5

Reform the equation by setting the left side equal to the right side.

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

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

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