Calculus Examples

$f (x) = 3 x \sqrt{x - x^{2}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

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

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

Step 1.1.2

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

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

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

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

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.7.2

Subtract $2$ from $1$ .

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

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

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

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

Multiply $2$ by $- 1$ .

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

Step 1.14

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

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

Step 1.15

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

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

Step 1.16

Simplify.

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Step 1.16.1

Apply the distributive property.

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

Step 1.16.2

Combine terms.

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Step 1.16.2.1

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

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

Step 1.16.2.2

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

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

Step 1.16.3

Reorder terms.

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

Step 1.16.4

Simplify each term.

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Step 1.16.4.1

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

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

Step 1.16.4.2

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

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

Step 1.16.5

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

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

Step 1.16.6

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

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

Step 1.16.7

Combine the numerators over the common denominator.

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

Step 1.16.8

Simplify the numerator.

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Step 1.16.8.1

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

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Step 1.16.8.1.1

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

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

Step 1.16.8.1.2

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

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

Step 1.16.8.1.3

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

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

Step 1.16.8.2

Apply the distributive property.

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

Step 1.16.8.3

Multiply $x$ by $1$ .

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

Step 1.16.8.4

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

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Step 1.16.8.4.1

Move $x$ .

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

Step 1.16.8.4.2

Multiply $x$ by $x$ .

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

Step 1.16.8.5

Rewrite using the commutative property of multiplication.

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

Step 1.16.8.6

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

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Step 1.16.8.6.1

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

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

Step 1.16.8.6.2

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

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

Step 1.16.8.6.3

Combine the numerators over the common denominator.

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

Step 1.16.8.6.4

Add $1$ and $1$ .

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

Step 1.16.8.6.5

Divide $2$ by $2$ .

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

Step 1.16.8.7

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

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

Step 1.16.8.8

Apply the distributive property.

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

Step 1.16.8.9

Multiply $- 1$ by $2$ .

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

Step 1.16.8.10

Add $x$ and $2 x$ .

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

Step 1.16.8.11

Subtract $2 x^{2}$ from $- 2 x^{2}$ .

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

Step 1.16.8.12

Factor $x$ out of $- 4 x^{2} + 3 x$ .

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Step 1.16.8.12.1

Factor $x$ out of $- 4 x^{2}$ .

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

Step 1.16.8.12.2

Factor $x$ out of $3 x$ .

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

Step 1.16.8.12.3

Factor $x$ out of $x (- 4 x) + x \cdot 3$ .

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

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

Step 1.16.9

Factor $- 1$ out of $- 4 x$ .

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

Step 1.16.10

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

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

Step 1.16.11

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

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

Step 1.16.12

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

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

Step 1.16.13

Move the negative in front of the fraction.

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

Step 1.16.14

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

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

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3 x (4 x - 3)}{2 {(x - x^{2})}^{\frac{1}{2}}}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [\frac{x (4 x - 3)}{{(x - x^{2})}^{\frac{1}{2}}}]$ .

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

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

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

Step 2.3.2.2

Rewrite the expression.

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

Step 2.4

Simplify.

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

Step 2.5

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 $g (x) = 4 x - 3$ .

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

Step 2.6

Differentiate.

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Step 2.6.1

By the Sum Rule, the derivative of $4 x - 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$ .

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

Step 2.6.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 2.6.3

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

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

Step 2.6.4

Multiply $4$ by $1$ .

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

Step 2.6.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 2.6.6

Simplify the expression.

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Step 2.6.6.1

Add $4$ and $0$ .

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

Step 2.6.6.2

Move $4$ to the left of $x$ .

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

Step 2.6.7

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

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

Step 2.6.8

Simplify by adding terms.

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Step 2.6.8.1

Multiply $4 x - 3$ by $1$ .

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

Step 2.6.8.2

Add $4 x$ and $4 x$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) \frac{d}{d x} {(x - x^{2})}^{\frac{1}{2}}}{x - x^{2}}$

Step 2.7

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

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Step 2.7.1

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.7.2

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.7.3

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{1}{2} \cdot {(x - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.8

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{1}{2} \cdot {(x - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.9

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{1}{2} \cdot {(x - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.10

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{1}{2} \cdot {(x - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{1}{2} \cdot {(x - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.11.2

Subtract $2$ from $1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{1}{2} \cdot {(x - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.12

Combine fractions.

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Step 2.12.1

Move the negative in front of the fraction.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{1}{2} \cdot {(x - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.12.2

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{{(x - x^{2})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.12.3

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - x (4 x - 3) (\frac{1}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x - x^{2}))}{x - x^{2}}$

Step 2.12.4

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot (4 x - 3) \frac{d}{d x} (x - x^{2})}{x - x^{2}}$

Step 2.13

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (\frac{d}{d x} (x) + \frac{d}{d x} (- x^{2})))}{x - x^{2}}$

Step 2.14

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 + \frac{d}{d x} (- x^{2})))}{x - x^{2}}$

Step 2.15

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - \frac{d}{d x} x^{2}))}{x - x^{2}}$

Step 2.16

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

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - (2 x)))}{x - x^{2}}$

Step 2.17

Combine fractions.

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Step 2.17.1

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x))}{x - x^{2}}$

Step 2.17.2

Multiply $\frac{{(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} (4 x - 3) (1 - 2 x)}{x - x^{2}}$ by $\frac{3}{2}$ .

$f'' (x) = - \frac{({(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x))) \cdot 3}{(x - x^{2}) \cdot 2}$

Step 2.17.3

Reorder.

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Step 2.17.3.1

Move $3$ to the left of ${(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} (4 x - 3) (1 - 2 x)$ .

$f'' (x) = - \frac{3 \cdot ({(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x)))}{(x - x^{2}) \cdot 2}$

Step 2.17.3.2

Move $2$ to the left of $x - x^{2}$ .

$f'' (x) = - \frac{3 ({(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x)))}{2 \cdot (x - x^{2})}$

Step 2.18

Simplify.

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Step 2.18.1

Apply the distributive property.

$f'' (x) = - \frac{3 ({(x - x^{2})}^{\frac{1}{2}} (8 x - 3)) + 3 (- \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x)))}{2 (x - x^{2})}$

Step 2.18.2

Apply the distributive property.

$f'' (x) = - \frac{3 ({(x - x^{2})}^{\frac{1}{2}} (8 x - 3)) + 3 (- \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x)))}{2 x + 2 (- x^{2})}$

Step 2.18.3

Simplify the numerator.

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Step 2.18.3.1

Factor $3$ out of $3 {(x - x^{2})}^{\frac{1}{2}} (8 x - 3) + 3 (- \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} (4 x - 3) (1 - 2 x))$ .

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Step 2.18.3.1.1

Factor $3$ out of $3 {(x - x^{2})}^{\frac{1}{2}} (8 x - 3)$ .

$f'' (x) = - \frac{3 ({(x - x^{2})}^{\frac{1}{2}} (8 x - 3)) + 3 (- \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x)))}{2 x + 2 (- x^{2})}$

Step 2.18.3.1.2

Factor $3$ out of $3 ({(x - x^{2})}^{\frac{1}{2}} (8 x - 3)) + 3 (- \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} (4 x - 3) (1 - 2 x))$ .

$f'' (x) = - \frac{3 ({(x - x^{2})}^{\frac{1}{2}} (8 x - 3) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x)))}{2 x + 2 (- x^{2})}$

Step 2.18.3.2

Apply the distributive property.

$f'' (x) = - \frac{3 ({(x - x^{2})}^{\frac{1}{2}} (8 x) + {(x - x^{2})}^{\frac{1}{2}} \cdot - 3 - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x)))}{2 x + 2 (- x^{2})}$

Step 2.18.3.3

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x + {(x - x^{2})}^{\frac{1}{2}} \cdot - 3 - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x)))}{2 x + 2 (- x^{2})}$

Step 2.18.3.4

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 \cdot {(x - x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot ((4 x - 3) (1 - 2 x)))}{2 x + 2 (- x^{2})}$

Step 2.18.3.5

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + (- \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot (4 x) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot - 3) (1 - 2 x))}{2 x + 2 (- x^{2})}$

Step 2.18.3.6

Cancel the common factor of $2$ .

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Step 2.18.3.6.1

Move the leading negative in $- \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}}$ into the numerator.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + (\frac{- x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot (4 x) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot - 3) (1 - 2 x))}{2 x + 2 (- x^{2})}$

Step 2.18.3.6.2

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + (\frac{- x}{2 ({(x - x^{2})}^{\frac{1}{2}})} \cdot (4 x) - \frac{x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot - 3) (1 - 2 x))}{2 x + 2 (- x^{2})}$

Step 2.18.3.6.3

Factor $2$ out of $4 x$ .

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

Step 2.18.3.6.4

Cancel the common factor.

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

Step 2.18.3.6.5

Rewrite the expression.

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

Step 2.18.3.7

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

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

Step 2.18.3.8

Multiply $- 1$ by $2$ .

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

Step 2.18.3.9

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

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

Step 2.18.3.10

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

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

Step 2.18.3.11

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

Step 2.18.3.12

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

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

Step 2.18.3.13

Add $1$ and $1$ .

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

Step 2.18.3.14

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

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Step 2.18.3.14.1

Multiply $- 3$ by $- 1$ .

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

Step 2.18.3.14.2

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

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

Step 2.18.3.15

Move the negative in front of the fraction.

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

Step 2.18.3.16

Expand $(- \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}}) (1 - 2 x)$ using the FOIL Method.

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Step 2.18.3.16.1

Apply the distributive property.

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

Step 2.18.3.16.2

Apply the distributive property.

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

Step 2.18.3.16.3

Apply the distributive property.

Step 2.18.3.17

Simplify and combine like terms.

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Step 2.18.3.17.1

Simplify each term.

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Step 2.18.3.17.1.1

Multiply $- 1$ by $1$ .

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

Step 2.18.3.17.1.2

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

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Step 2.18.3.17.1.2.1

Multiply $- 2$ by $- 1$ .

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

Step 2.18.3.17.1.2.2

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

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

Step 2.18.3.17.1.2.3

Multiply $2$ by $2$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} \cdot x + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot 1 + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x))}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.2.4

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{2} x}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot 1 + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x))}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.2.5

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 (x \cdot x^{2})}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot 1 + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x))}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.2.6

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{1 + 2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot 1 + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x))}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.2.7

Add $1$ and $2$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot 1 + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x))}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.3

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot (- 2 x))}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} - 2 \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} \cdot x)}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.5

Cancel the common factor of $2$ .

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Step 2.18.3.17.1.5.1

Factor $2$ out of $- 2$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} + 2 \cdot (- 1 \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}}) \cdot x)}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.5.2

Cancel the common factor.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} + 2 \cdot (- 1 \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}}) \cdot x)}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.5.3

Rewrite the expression.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} - \frac{3 x}{{(x - x^{2})}^{\frac{1}{2}}} \cdot x)}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.6

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

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Step 2.18.3.17.1.6.1

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} - \frac{x (3 x)}{{(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.6.2

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} - \frac{3 (x \cdot x)}{{(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.6.3

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} - \frac{3 (x \cdot x)}{{(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.6.4

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} - \frac{3 x^{1 + 1}}{{(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.1.6.5

Add $1$ and $1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} - \frac{2 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} - \frac{3 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.17.2

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{4 x^{3}}{{(x - x^{2})}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}} + \frac{- 2 x^{2} - 3 x^{2}}{{(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.18

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{4 x^{3} - 2 x^{2} - 3 x^{2}}{{(- x^{2} + x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.19

Subtract $3 x^{2}$ from $- 2 x^{2}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{4 x^{3} - 5 x^{2}}{{(- x^{2} + x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.20

Factor $x^{2}$ out of $4 x^{3} - 5 x^{2}$ .

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Step 2.18.3.20.1

Factor $x^{2}$ out of $4 x^{3}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x^{2} (4 x) - 5 x^{2}}{{(- x^{2} + x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.20.2

Factor $x^{2}$ out of $- 5 x^{2}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x^{2} (4 x) + x^{2} \cdot - 5}{{(- x^{2} + x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.20.3

Factor $x^{2}$ out of $x^{2} (4 x) + x^{2} \cdot - 5$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x^{2} (4 x - 5)}{{(- x^{2} + x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(x - x^{2})}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.21

Reorder terms.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x^{2} (4 x - 5)}{{(- x^{2} + x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.22

To write $\frac{x^{2} (4 x - 5)}{{(- x^{2} + x)}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x^{2} (4 x - 5)}{{(- x^{2} + x)}^{\frac{1}{2}}} \cdot \frac{2}{2} + \frac{3 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.23

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

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Step 2.18.3.23.1

Multiply $\frac{x^{2} (4 x - 5)}{{(- x^{2} + x)}^{\frac{1}{2}}}$ by $\frac{2}{2}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x^{2} (4 x - 5) \cdot 2}{{(- x^{2} + x)}^{\frac{1}{2}} \cdot 2} + \frac{3 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.23.2

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

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x^{2} (4 x - 5) \cdot 2}{2 {(- x^{2} + x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.24

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x^{2} (4 x - 5) \cdot 2 + 3 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25

Simplify the numerator.

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Step 2.18.3.25.1

Factor $x$ out of $x^{2} (4 x - 5) \cdot 2 + 3 x$ .

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Step 2.18.3.25.1.1

Factor $x$ out of $x^{2} (4 x - 5) \cdot 2$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((x (4 x - 5)) \cdot 2) + 3 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.1.2

Factor $x$ out of $3 x$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((x (4 x - 5)) \cdot 2) + x \cdot 3}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.1.3

Factor $x$ out of $x ((x (4 x - 5)) \cdot 2) + x \cdot 3$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((x (4 x - 5)) \cdot 2 + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.2

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((x (4 x) + x \cdot - 5) \cdot 2 + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.3

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((4 x \cdot x + x \cdot - 5) \cdot 2 + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.4

Move $- 5$ to the left of $x$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((4 x \cdot x - 5 \cdot x) \cdot 2 + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.5

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

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Step 2.18.3.25.5.1

Move $x$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((4 (x \cdot x) - 5 \cdot x) \cdot 2 + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.5.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((4 x^{2} - 5 \cdot x) \cdot 2 + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((4 x^{2} - 5 x) \cdot 2 + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.6

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x (4 x^{2} \cdot 2 - 5 x \cdot 2 + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.7

Multiply $2$ by $4$ .

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

Step 2.18.3.25.8

Multiply $2$ by $- 5$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x (8 x^{2} - 10 x + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.9

Factor by grouping.

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Step 2.18.3.25.9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 8 \cdot 3 = 24$ and whose sum is $b = - 10$ .

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Step 2.18.3.25.9.1.1

Factor $- 10$ out of $- 10 x$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x (8 x^{2} - 10 x + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.9.1.2

Rewrite $- 10$ as $- 4$ plus $- 6$

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x (8 x^{2} + (- 4 - 6) x + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.9.1.3

Apply the distributive property.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x (8 x^{2} - 4 x - 6 x + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.9.2

Factor out the greatest common factor from each group.

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Step 2.18.3.25.9.2.1

Group the first two terms and the last two terms.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((8 x^{2} - 4 x) - 6 x + 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.9.2.2

Factor out the greatest common factor (GCF) from each group.

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x (4 x (2 x - 1) - 3 (2 x - 1))}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.25.9.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x ((2 x - 1) (4 x - 3))}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x - 3 {(x - x^{2})}^{\frac{1}{2}} + \frac{x (2 x - 1) (4 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.26

To write $8 {(x - x^{2})}^{\frac{1}{2}} x$ as a fraction with a common denominator, multiply by $\frac{2 {(- x^{2} + x)}^{\frac{1}{2}}}{2 {(- x^{2} + x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 (8 {(x - x^{2})}^{\frac{1}{2}} x \cdot \frac{2 {(- x^{2} + x)}^{\frac{1}{2}}}{2 {(- x^{2} + x)}^{\frac{1}{2}}} + \frac{x (2 x - 1) (4 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}} - 3 {(x - x^{2})}^{\frac{1}{2}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.27

Combine $8 {(x - x^{2})}^{\frac{1}{2}} x$ and $\frac{2 {(- x^{2} + x)}^{\frac{1}{2}}}{2 {(- x^{2} + x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 (\frac{8 {(x - x^{2})}^{\frac{1}{2}} x (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}} + \frac{x (2 x - 1) (4 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}} - 3 {(x - x^{2})}^{\frac{1}{2}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.28

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (\frac{8 {(x - x^{2})}^{\frac{1}{2}} x (2 {(- x^{2} + x)}^{\frac{1}{2}}) + x (2 x - 1) (4 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}} - 3 {(x - x^{2})}^{\frac{1}{2}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.29

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

Step 2.18.3.30

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

$f'' (x) = - \frac{3 (\frac{8 {(x - x^{2})}^{\frac{1}{2}} x (2 {(- x^{2} + x)}^{\frac{1}{2}}) + x (2 x - 1) (4 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}} + \frac{- 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.31

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (\frac{8 {(x - x^{2})}^{\frac{1}{2}} x (2 {(- x^{2} + x)}^{\frac{1}{2}}) + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32

Rewrite $\frac{8 {(x - x^{2})}^{\frac{1}{2}} x (2 {(- x^{2} + x)}^{\frac{1}{2}}) + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}}$ in a factored form.

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Step 2.18.3.32.1

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

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Step 2.18.3.32.1.1

Multiply $2$ by $8$ .

$f'' (x) = - \frac{3 (\frac{16 {(x - x^{2})}^{\frac{1}{2}} x {(- x^{2} + x)}^{\frac{1}{2}} + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.1.2

Reorder terms.

$f'' (x) = - \frac{3 (\frac{16 {(- x^{2} + x)}^{\frac{1}{2}} x {(- x^{2} + x)}^{\frac{1}{2}} + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.1.3

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

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Step 2.18.3.32.1.3.1

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

$f'' (x) = - \frac{3 (\frac{16 ({(- x^{2} + x)}^{\frac{1}{2}} {(- x^{2} + x)}^{\frac{1}{2}}) x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.1.3.2

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

$f'' (x) = - \frac{3 (\frac{16 {(- x^{2} + x)}^{\frac{1}{2} + \frac{1}{2}} x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.1.3.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (\frac{16 {(- x^{2} + x)}^{\frac{1 + 1}{2}} x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.1.3.4

Add $1$ and $1$ .

$f'' (x) = - \frac{3 (\frac{16 {(- x^{2} + x)}^{\frac{2}{2}} x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.1.3.5

Divide $2$ by $2$ .

$f'' (x) = - \frac{3 (\frac{16 (- x^{2} + x) x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.1.4

Simplify $16 {(- x^{2} + x)}^{1} x$ .

$f'' (x) = - \frac{3 (\frac{16 (- x^{2} + x) x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.2

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{(16 (- x^{2}) + 16 x) x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.3

Multiply $- 1$ by $16$ .

$f'' (x) = - \frac{3 (\frac{(- 16 x^{2} + 16 x) x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.4

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{2} x + 16 x \cdot x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.5

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

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Step 2.18.3.32.5.1

Move $x$ .

$f'' (x) = - \frac{3 (\frac{- 16 (x \cdot x^{2}) + 16 x \cdot x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.5.2

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

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Step 2.18.3.32.5.2.1

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

Step 2.18.3.32.5.2.2

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

$f'' (x) = - \frac{3 (\frac{- 16 x^{1 + 2} + 16 x \cdot x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.5.3

Add $1$ and $2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x \cdot x + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.6

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

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Step 2.18.3.32.6.1

Move $x$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 (x \cdot x) + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.6.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + x (2 x - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.7

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + (x (2 x) + x \cdot - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.8

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + (2 x \cdot x + x \cdot - 1) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.9

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

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + (2 x \cdot x - 1 \cdot x) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.10

Simplify each term.

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Step 2.18.3.32.10.1

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

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Step 2.18.3.32.10.1.1

Move $x$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + (2 (x \cdot x) - 1 \cdot x) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.10.1.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + (2 x^{2} - 1 \cdot x) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.10.2

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

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + (2 x^{2} - x) (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.11

Expand $(2 x^{2} - x) (4 x - 3)$ using the FOIL Method.

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Step 2.18.3.32.11.1

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 2 x^{2} (4 x - 3) - x (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.11.2

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 2 x^{2} (4 x) + 2 x^{2} \cdot - 3 - x (4 x - 3) - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.11.3

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 2 x^{2} (4 x) + 2 x^{2} \cdot - 3 - x (4 x) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12

Simplify and combine like terms.

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Step 2.18.3.32.12.1

Simplify each term.

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Step 2.18.3.32.12.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 2 \cdot (4 x^{2} x) + 2 x^{2} \cdot - 3 - x (4 x) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.2

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

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Step 2.18.3.32.12.1.2.1

Move $x$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 2 \cdot (4 (x \cdot x^{2})) + 2 x^{2} \cdot - 3 - x (4 x) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.2.2

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

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Step 2.18.3.32.12.1.2.2.1

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

Step 2.18.3.32.12.1.2.2.2

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

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 2 \cdot (4 x^{1 + 2}) + 2 x^{2} \cdot - 3 - x (4 x) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.2.3

Add $1$ and $2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 2 \cdot (4 x^{3}) + 2 x^{2} \cdot - 3 - x (4 x) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.3

Multiply $2$ by $4$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} + 2 x^{2} \cdot - 3 - x (4 x) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.4

Multiply $- 3$ by $2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 6 x^{2} - x (4 x) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.5

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 6 x^{2} - 1 \cdot (4 x \cdot x) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.6

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

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Step 2.18.3.32.12.1.6.1

Move $x$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 6 x^{2} - 1 \cdot (4 (x \cdot x)) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.6.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 6 x^{2} - 1 \cdot (4 x^{2}) - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.7

Multiply $- 1$ by $4$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 6 x^{2} - 4 x^{2} - x \cdot - 3 - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.1.8

Multiply $- 3$ by $- 1$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 6 x^{2} - 4 x^{2} + 3 x - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.12.2

Subtract $4 x^{2}$ from $- 6 x^{2}$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 3 {(x - x^{2})}^{\frac{1}{2}} (2 {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.13

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 3 \cdot (2 {(x - x^{2})}^{\frac{1}{2}} {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.14

Multiply $- 3$ by $2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 6 {(x - x^{2})}^{\frac{1}{2}} {(- x^{2} + x)}^{\frac{1}{2}}}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.15

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

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Step 2.18.3.32.15.1

Reorder terms.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 6 {(- x^{2} + x)}^{\frac{1}{2}} {(- x^{2} + x)}^{\frac{1}{2}}}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.15.2

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

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Step 2.18.3.32.15.2.1

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

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 6 ({(- x^{2} + x)}^{\frac{1}{2}} {(- x^{2} + x)}^{\frac{1}{2}})}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.15.2.2

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

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 6 {(- x^{2} + x)}^{\frac{1}{2} + \frac{1}{2}}}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.15.2.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 6 {(- x^{2} + x)}^{\frac{1 + 1}{2}}}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.15.2.4

Add $1$ and $1$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 6 {(- x^{2} + x)}^{\frac{2}{2}}}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.15.2.5

Divide $2$ by $2$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 6 (- x^{2} + x)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.15.3

Simplify $- 6 {(- x^{2} + x)}^{1}$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 6 (- x^{2} + x)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.16

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x - 6 (- x^{2}) - 6 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.17

Multiply $- 1$ by $- 6$ .

$f'' (x) = - \frac{3 (\frac{- 16 x^{3} + 16 x^{2} + 8 x^{3} - 10 x^{2} + 3 x + 6 x^{2} - 6 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.18

Add $- 16 x^{3}$ and $8 x^{3}$ .

$f'' (x) = - \frac{3 (\frac{- 8 x^{3} + 16 x^{2} - 10 x^{2} + 3 x + 6 x^{2} - 6 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.19

Subtract $10 x^{2}$ from $16 x^{2}$ .

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

Step 2.18.3.32.20

Add $6 x^{2}$ and $6 x^{2}$ .

$f'' (x) = - \frac{3 (\frac{- 8 x^{3} + 12 x^{2} + 3 x - 6 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.21

Subtract $6 x$ from $3 x$ .

$f'' (x) = - \frac{3 (\frac{- 8 x^{3} + 12 x^{2} - 3 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.22

Factor $x$ out of $- 8 x^{3} + 12 x^{2} - 3 x$ .

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Step 2.18.3.32.22.1

Factor $x$ out of $- 8 x^{3}$ .

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{2}) + 12 x^{2} - 3 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.22.2

Factor $x$ out of $12 x^{2}$ .

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{2}) + x (12 x) - 3 x}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.22.3

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

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{2}) + x (12 x) + x \cdot - 3}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.22.4

Factor $x$ out of $x (- 8 x^{2}) + x (12 x)$ .

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{2} + 12 x) + x \cdot - 3}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.3.32.22.5

Factor $x$ out of $x (- 8 x^{2} + 12 x) + x \cdot - 3$ .

$f'' (x) = - \frac{3 (\frac{x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}})}{2 x + 2 (- x^{2})}$

Step 2.18.4

Combine terms.

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Step 2.18.4.1

Combine $3$ and $\frac{x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{\frac{3 (x (- 8 x^{2} + 12 x - 3))}{2 {(- x^{2} + x)}^{\frac{1}{2}}}}{2 x + 2 (- x^{2})}$

Step 2.18.4.2

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{\frac{3 x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}}}{2 x - 2 x^{2}}$

Step 2.18.4.3

Rewrite $\frac{\frac{3 x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}}}{2 x - 2 x^{2}}$ as a product.

$f'' (x) = - (\frac{3 x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}} \cdot \frac{1}{2 x - 2 x^{2}})$

Step 2.18.4.4

Multiply $\frac{3 x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}}}$ by $\frac{1}{2 x - 2 x^{2}}$ .

$f'' (x) = - \frac{3 x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}} (2 x - 2 x^{2})}$

Step 2.18.5

Simplify the denominator.

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Step 2.18.5.1

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

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Step 2.18.5.1.1

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

$f'' (x) = - \frac{3 x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}} (2 x (1) - 2 x^{2})}$

Step 2.18.5.1.2

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

$f'' (x) = - \frac{3 x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}} (2 x (1) + 2 x (- x))}$

Step 2.18.5.1.3

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

$f'' (x) = - \frac{3 x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}} (2 x (1 - x))}$

$f'' (x) = - \frac{3 x (- 8 x^{2} + 12 x - 3)}{2 {(- x^{2} + x)}^{\frac{1}{2}} \cdot (2 x (1 - x))}$

Step 2.18.5.2

Multiply $2$ by $2$ .

$f'' (x) = - \frac{3 x (- 8 x^{2} + 12 x - 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} x (1 - x)}$

Step 2.18.6

Cancel the common factor.

$f'' (x) = - \frac{3 x (- 8 x^{2} + 12 x - 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} x (1 - x)}$

Step 2.18.7

Rewrite the expression.

$f'' (x) = - \frac{3 (- 8 x^{2} + 12 x - 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$

Step 2.18.8

Factor $- 1$ out of $- 8 x^{2}$ .

$f'' (x) = - \frac{3 (- (8 x^{2}) + 12 x - 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$

Step 2.18.9

Factor $- 1$ out of $12 x$ .

$f'' (x) = - \frac{3 (- (8 x^{2}) - (- 12 x) - 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$

Step 2.18.10

Factor $- 1$ out of $- (8 x^{2}) - (- 12 x)$ .

$f'' (x) = - \frac{3 (- (8 x^{2} - 12 x) - 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$

Step 2.18.11

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

$f'' (x) = - \frac{3 (- (8 x^{2} - 12 x) - 1 \cdot 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$

Step 2.18.12

Factor $- 1$ out of $- (8 x^{2} - 12 x) - 1 (3)$ .

$f'' (x) = - \frac{3 (- (8 x^{2} - 12 x + 3))}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$

Step 2.18.13

Rewrite $- (8 x^{2} - 12 x + 3)$ as $- 1 (8 x^{2} - 12 x + 3)$ .

$f'' (x) = - \frac{3 (- 1 (8 x^{2} - 12 x + 3))}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$

Step 2.18.14

Move the negative in front of the fraction.

$f'' (x) = \frac{3 (8 x^{2} - 12 x + 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$

Step 2.18.15

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{3 (8 x^{2} - 12 x + 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)})$

Step 2.18.16

Multiply $\frac{3 (8 x^{2} - 12 x + 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$ by $1$ .

$f'' (x) = \frac{3 (8 x^{2} - 12 x + 3)}{4 {(- x^{2} + x)}^{\frac{1}{2}} (1 - x)}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

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

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.2

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

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

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

Step 4.1.5

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

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

Step 4.1.6

Combine the numerators over the common denominator.

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

Step 4.1.7

Simplify the numerator.

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Step 4.1.7.1

Multiply $- 1$ by $2$ .

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

Step 4.1.7.2

Subtract $2$ from $1$ .

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

Step 4.1.8

Combine fractions.

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Step 4.1.8.1

Move the negative in front of the fraction.

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

Step 4.1.8.2

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

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

Step 4.1.8.3

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

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

Step 4.1.8.4

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

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

Step 4.1.9

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

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

Step 4.1.10

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

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

Step 4.1.11

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

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

Step 4.1.12

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

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

Step 4.1.13

Multiply $2$ by $- 1$ .

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

Step 4.1.14

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

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

Step 4.1.15

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

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

Step 4.1.16

Simplify.

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Step 4.1.16.1

Apply the distributive property.

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

Step 4.1.16.2

Combine terms.

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Step 4.1.16.2.1

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

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

Step 4.1.16.2.2

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

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

Step 4.1.16.3

Reorder terms.

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

Step 4.1.16.4

Simplify each term.

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Step 4.1.16.4.1

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

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

Step 4.1.16.4.2

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

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

Step 4.1.16.5

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

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

Step 4.1.16.6

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

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

Step 4.1.16.7

Combine the numerators over the common denominator.

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

Step 4.1.16.8

Simplify the numerator.

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Step 4.1.16.8.1

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

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Step 4.1.16.8.1.1

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

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

Step 4.1.16.8.1.2

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

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

Step 4.1.16.8.1.3

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

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

Step 4.1.16.8.2

Apply the distributive property.

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

Step 4.1.16.8.3

Multiply $x$ by $1$ .

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

Step 4.1.16.8.4

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

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Step 4.1.16.8.4.1

Move $x$ .

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

Step 4.1.16.8.4.2

Multiply $x$ by $x$ .

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

Step 4.1.16.8.5

Rewrite using the commutative property of multiplication.

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

Step 4.1.16.8.6

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

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Step 4.1.16.8.6.1

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

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

Step 4.1.16.8.6.2

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

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

Step 4.1.16.8.6.3

Combine the numerators over the common denominator.

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

Step 4.1.16.8.6.4

Add $1$ and $1$ .

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

Step 4.1.16.8.6.5

Divide $2$ by $2$ .

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

Step 4.1.16.8.7

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

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

Step 4.1.16.8.8

Apply the distributive property.

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

Step 4.1.16.8.9

Multiply $- 1$ by $2$ .

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

Step 4.1.16.8.10

Add $x$ and $2 x$ .

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

Step 4.1.16.8.11

Subtract $2 x^{2}$ from $- 2 x^{2}$ .

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

Step 4.1.16.8.12

Factor $x$ out of $- 4 x^{2} + 3 x$ .

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Step 4.1.16.8.12.1

Factor $x$ out of $- 4 x^{2}$ .

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

Step 4.1.16.8.12.2

Factor $x$ out of $3 x$ .

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

Step 4.1.16.8.12.3

Factor $x$ out of $x (- 4 x) + x \cdot 3$ .

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

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

Step 4.1.16.9

Factor $- 1$ out of $- 4 x$ .

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

Step 4.1.16.10

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

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

Step 4.1.16.11

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

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

Step 4.1.16.12

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

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

Step 4.1.16.13

Move the negative in front of the fraction.

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

Step 4.1.16.14

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

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{3 x (4 x - 3)}{2 {(x - x^{2})}^{\frac{1}{2}}}$ .

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

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{3 x (4 x - 3)}{2 {(x - x^{2})}^{\frac{1}{2}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Set the numerator equal to zero.

$3 x (4 x - 3) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$4 x - 3 = 0$

Step 5.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 5.3.3

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 5.3.3.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 5.3.3.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 5.3.3.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 5.3.3.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 5.3.3.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 5.3.3.2.2.2

Simplify the left side.

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Step 5.3.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 5.3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 5.3.4

The final solution is all the values that make $3 x (4 x - 3) = 0$ true.

$x = 0, \frac{3}{4}$

Step 5.4

Exclude the solutions that do not make $- \frac{3 x (4 x - 3)}{2 {(x - x^{2})}^{\frac{1}{2}}} = 0$ true.

$x = \frac{3}{4}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Convert expressions with fractional exponents to radicals.

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Step 6.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 6.1.2

Anything raised to $1$ is the base itself.

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

Step 6.2

Set the denominator in $\frac{3 x (4 x - 3)}{2 \sqrt{x - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x - x^{2}} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{x - x^{2}})}^{2} = 0^{2}$

Step 6.3.2

Simplify each side of the equation.

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Step 6.3.2.1

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

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

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Step 6.3.2.2.1.1

Apply the product rule to $2 {(x - x^{2})}^{\frac{1}{2}}$ .

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

Step 6.3.2.2.1.2

Raise $2$ to the power of $2$ .

$4 {({(x - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.3.2.2.1.3

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

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Step 6.3.2.2.1.3.1

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

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

Step 6.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1.3.2.1

Cancel the common factor.

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

Step 6.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(x - x^{2})}^{1} = 0^{2}$

Step 6.3.2.2.1.4

Simplify.

$4 (x - x^{2}) = 0^{2}$

Step 6.3.2.2.1.5

Apply the distributive property.

$4 x + 4 (- x^{2}) = 0^{2}$

Step 6.3.2.2.1.6

Multiply $- 1$ by $4$ .

$4 x - 4 x^{2} = 0^{2}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$4 x - 4 x^{2} = 0$

Step 6.3.3

Solve for $x$ .

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Step 6.3.3.1

Factor the left side of the equation.

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Step 6.3.3.1.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$4 u - 4 u^{2} = 0$

Step 6.3.3.1.2

Factor $4 u$ out of $4 u - 4 u^{2}$ .

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Step 6.3.3.1.2.1

Factor $4 u$ out of $4 u$ .

$4 u (1) - 4 u^{2} = 0$

Step 6.3.3.1.2.2

Factor $4 u$ out of $- 4 u^{2}$ .

$4 u (1) + 4 u (- u) = 0$

Step 6.3.3.1.2.3

Factor $4 u$ out of $4 u (1) + 4 u (- u)$ .

$4 u (1 - u) = 0$

Step 6.3.3.1.3

Replace all occurrences of $u$ with $x$ .

$4 x (1 - x) = 0$

Step 6.3.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 - x = 0$

Step 6.3.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 6.3.3.4

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 6.3.3.4.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 6.3.3.4.2

Solve $1 - x = 0$ for $x$ .

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Step 6.3.3.4.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 6.3.3.4.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 6.3.3.4.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 6.3.3.4.2.2.2

Simplify the left side.

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Step 6.3.3.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 6.3.3.4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 6.3.3.4.2.2.3

Simplify the right side.

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Step 6.3.3.4.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 6.3.3.5

The final solution is all the values that make $4 x (1 - x) = 0$ true.

$x = 0, 1$

Step 6.4

Set the radicand in $\sqrt{x - x^{2}}$ less than $0$ to find where the expression is undefined.

$x - x^{2} < 0$

Step 6.5

Solve for $x$ .

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Step 6.5.1

Convert the inequality to an equation.

$x - x^{2} = 0$

Step 6.5.2

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

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Step 6.5.2.1

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - x^{2} = 0$

Step 6.5.2.2

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

$x \cdot 1 + x (- x) = 0$

Step 6.5.2.3

Factor $x$ out of $x \cdot 1 + x (- x)$ .

$x (1 - x) = 0$

Step 6.5.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 - x = 0$

Step 6.5.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.5.5

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 6.5.5.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 6.5.5.2

Solve $1 - x = 0$ for $x$ .

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Step 6.5.5.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 6.5.5.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 6.5.5.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 6.5.5.2.2.2

Simplify the left side.

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Step 6.5.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 6.5.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 6.5.5.2.2.3

Simplify the right side.

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Step 6.5.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 6.5.6

The final solution is all the values that make $x (1 - x) = 0$ true.

$x = 0, 1$

Step 6.5.7

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 6.5.8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 6.5.8.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 6.5.8.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 6.5.8.1.2

Replace $x$ with $- 2$ in the original inequality.

$(- 2) - {(- 2)}^{2} < 0$

Step 6.5.8.1.3

The left side $- 6$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.5.8.2

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

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Step 6.5.8.2.1

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Step 6.5.8.2.2

Replace $x$ with $0.5$ in the original inequality.

$(0.5) - {(0.5)}^{2} < 0$

Step 6.5.8.2.3

The left side $0.25$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 6.5.8.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 6.5.8.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 6.5.8.3.2

Replace $x$ with $4$ in the original inequality.

$(4) - {(4)}^{2} < 0$

Step 6.5.8.3.3

The left side $- 12$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.5.8.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ True

$0 < x < 1$ False

$x > 1$ True

$x < 0$ True

$0 < x < 1$ False

$x > 1$ True

Step 6.5.9

The solution consists of all of the true intervals.

$x < 0$ or $x > 1$

Step 6.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0, x \geq 1$

$(- \infty, 0] \cup [1, \infty)$

$x \leq 0, x \geq 1$

$(- \infty, 0] \cup [1, \infty)$

Step 7

Critical points to evaluate.

$x = \frac{3}{4}, 0, 1$

Step 8

Evaluate the second derivative at $x = \frac{3}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3 (8 {(\frac{3}{4})}^{2} - 12 (\frac{3}{4}) + 3)}{4 {(- {(\frac{3}{4})}^{2} + \frac{3}{4})}^{\frac{1}{2}} (1 - (\frac{3}{4}))}$

Step 9

Evaluate the second derivative.

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Step 9.1

Remove parentheses.

$\frac{3 (8 {(\frac{3}{4})}^{2} - 12 (\frac{3}{4}) + 3)}{4 {(- {(\frac{3}{4})}^{2} + \frac{3}{4})}^{\frac{1}{2}} (1 - (\frac{3}{4}))}$

Step 9.2

Simplify the numerator.

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Step 9.2.1

Apply the product rule to $\frac{3}{4}$ .

$\frac{3 (8 \frac{3^{2}}{4^{2}} - 12 (\frac{3}{4}) + 3)}{4 {(- {(\frac{3}{4})}^{2} + \frac{3}{4})}^{\frac{1}{2}} (1 - \frac{3}{4})}$

Step 9.2.2

Raise $3$ to the power of $2$ .

$\frac{3 (8 \frac{9}{4^{2}} - 12 (\frac{3}{4}) + 3)}{4 {(- {(\frac{3}{4})}^{2} + \frac{3}{4})}^{\frac{1}{2}} (1 - \frac{3}{4})}$

Step 9.2.3

Raise $4$ to the power of $2$ .

$\frac{3 (8 (\frac{9}{16}) - 12 (\frac{3}{4}) + 3)}{4 {(- {(\frac{3}{4})}^{2} + \frac{3}{4})}^{\frac{1}{2}} (1 - \frac{3}{4})}$

Step 9.2.4

Cancel the common factor of $8$ .

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Step 9.2.4.1

Factor $8$ out of $16$ .

$\frac{3 (8 \frac{9}{8 (2)} - 12 (\frac{3}{4}) + 3)}{4 {(- {(\frac{3}{4})}^{2} + \frac{3}{4})}^{\frac{1}{2}} (1 - \frac{3}{4})}$

Step 9.2.4.2

Cancel the common factor.

$\frac{3 (8 \frac{9}{8 \cdot 2} - 12 (\frac{3}{4}) + 3)}{4 {(- {(\frac{3}{4})}^{2} + \frac{3}{4})}^{\frac{1}{2}} (1 - \frac{3}{4})}$

Step 9.2.4.3

Rewrite the expression.

$\frac{3 (\frac{9}{2} - 12 (\frac{3}{4}) + 3)}{4 {(- {(\frac{3}{4})}^{2} + \frac{3}{4})}^{\frac{1}{2}} (1 - \frac{3}{4})}$

Step 9.2.5

Cancel the common factor of $4$ .

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Step 9.2.5.1

Factor $4$ out of $- 12$ .

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

Step 9.2.5.2

Cancel the common factor.

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

Step 9.2.5.3

Rewrite the expression.

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

Step 9.2.6

Multiply $- 3$ by $3$ .

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

Step 9.2.7

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9.2.8

Combine $- 9$ and $\frac{2}{2}$ .

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

Step 9.2.9

Combine the numerators over the common denominator.

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

Step 9.2.10

Simplify the numerator.

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Step 9.2.10.1

Multiply $- 9$ by $2$ .

$\frac{3 (\frac{9 - 18}{2} + 3)}{4 {(- {(\frac{3}{4})}^{2} + \frac{3}{4})}^{\frac{1}{2}} (1 - \frac{3}{4})}$

Step 9.2.10.2

Subtract $18$ from $9$ .

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

Step 9.2.11

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

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

Step 9.2.12

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

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

Step 9.2.13

Combine the numerators over the common denominator.

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

Step 9.2.14

Simplify the numerator.

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Step 9.2.14.1

Multiply $3$ by $2$ .

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

Step 9.2.14.2

Add $- 9$ and $6$ .

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

Step 9.2.15

Move the negative in front of the fraction.

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

Step 9.2.16

Combine exponents.

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Step 9.2.16.1

Factor out negative.

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

Step 9.2.16.2

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

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

Step 9.2.16.3

Multiply $3$ by $3$ .

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

Step 9.3

Simplify the denominator.

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Step 9.3.1

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

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

Step 9.3.2

Combine the numerators over the common denominator.

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

Step 9.3.3

Subtract $3$ from $4$ .

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

Step 9.3.4

Combine exponents.

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Step 9.3.4.1

Combine $\frac{1}{4}$ and $4$ .

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

Step 9.3.4.2

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

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

Step 9.3.5

Cancel the common factor.

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

Step 9.3.6

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

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

Step 9.3.7

Simplify each term.

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Step 9.3.7.1

Apply the product rule to $\frac{3}{4}$ .

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

Step 9.3.7.2

Raise $3$ to the power of $2$ .

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

Step 9.3.7.3

Raise $4$ to the power of $2$ .

$\frac{- \frac{9}{2}}{{(- \frac{9}{16} + \frac{3}{4})}^{\frac{1}{2}}}$

Step 9.3.8

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{- \frac{9}{2}}{{(- \frac{9}{16} + \frac{3}{4} \cdot \frac{4}{4})}^{\frac{1}{2}}}$

Step 9.3.9

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.9.1

Multiply $\frac{3}{4}$ by $\frac{4}{4}$ .

$\frac{- \frac{9}{2}}{{(- \frac{9}{16} + \frac{3 \cdot 4}{4 \cdot 4})}^{\frac{1}{2}}}$

Step 9.3.9.2

Multiply $4$ by $4$ .

$\frac{- \frac{9}{2}}{{(- \frac{9}{16} + \frac{3 \cdot 4}{16})}^{\frac{1}{2}}}$

Step 9.3.10

Combine the numerators over the common denominator.

$\frac{- \frac{9}{2}}{{(\frac{- 9 + 3 \cdot 4}{16})}^{\frac{1}{2}}}$

Step 9.3.11

Simplify the numerator.

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Step 9.3.11.1

Multiply $3$ by $4$ .

$\frac{- \frac{9}{2}}{{(\frac{- 9 + 12}{16})}^{\frac{1}{2}}}$

Step 9.3.11.2

Add $- 9$ and $12$ .

$\frac{- \frac{9}{2}}{{(\frac{3}{16})}^{\frac{1}{2}}}$

Step 9.3.12

Apply the product rule to $\frac{3}{16}$ .

$\frac{- \frac{9}{2}}{\frac{3^{\frac{1}{2}}}{16^{\frac{1}{2}}}}$

Step 9.3.13

Simplify the denominator.

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Step 9.3.13.1

Rewrite $16$ as $4^{2}$ .

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

Step 9.3.13.2

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

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

Step 9.3.13.3

Cancel the common factor of $2$ .

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Step 9.3.13.3.1

Cancel the common factor.

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

Step 9.3.13.3.2

Rewrite the expression.

$\frac{- \frac{9}{2}}{\frac{3^{\frac{1}{2}}}{4^{1}}}$

Step 9.3.13.4

Evaluate the exponent.

$\frac{- \frac{9}{2}}{\frac{3^{\frac{1}{2}}}{4}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$- \frac{9}{2} \cdot \frac{4}{3^{\frac{1}{2}}}$

Step 9.5

Cancel the common factor of $2$ .

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Step 9.5.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$\frac{- 9}{2} \cdot \frac{4}{3^{\frac{1}{2}}}$

Step 9.5.2

Factor $2$ out of $4$ .

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

Step 9.5.3

Cancel the common factor.

$\frac{- 9}{2} \cdot \frac{2 \cdot 2}{3^{\frac{1}{2}}}$

Step 9.5.4

Rewrite the expression.

$- 9 \frac{2}{3^{\frac{1}{2}}}$

Step 9.6

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

$\frac{- 9 \cdot 2}{3^{\frac{1}{2}}}$

Step 9.7

Simplify the expression.

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Step 9.7.1

Multiply $- 9$ by $2$ .

$\frac{- 18}{3^{\frac{1}{2}}}$

Step 9.7.2

Move the negative in front of the fraction.

$- \frac{18}{3^{\frac{1}{2}}}$

Step 10

$x = \frac{3}{4}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{3}{4}$ is a local maximum

Step 11

Find the y-value when $x = \frac{3}{4}$ .

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Step 11.1

Replace the variable $x$ with $\frac{3}{4}$ in the expression.

$f (\frac{3}{4}) = 3 (\frac{3}{4}) \sqrt{(\frac{3}{4}) - {(\frac{3}{4})}^{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Multiply $3 (\frac{3}{4})$ .

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Step 11.2.1.1

Combine $3$ and $\frac{3}{4}$ .

$f (\frac{3}{4}) = \frac{3 \cdot 3}{4} \cdot \sqrt{(\frac{3}{4}) - {(\frac{3}{4})}^{2}}$

Step 11.2.1.2

Multiply $3$ by $3$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{(\frac{3}{4}) - {(\frac{3}{4})}^{2}}$

Step 11.2.2

Apply the product rule to $\frac{3}{4}$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{\frac{3}{4} - \frac{3^{2}}{4^{2}}}$

Step 11.2.3

Raise $3$ to the power of $2$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{\frac{3}{4} - \frac{9}{4^{2}}}$

Step 11.2.4

Raise $4$ to the power of $2$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{\frac{3}{4} - \frac{9}{16}}$

Step 11.2.5

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{\frac{3}{4} \cdot \frac{4}{4} - \frac{9}{16}}$

Step 11.2.6

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.6.1

Multiply $\frac{3}{4}$ by $\frac{4}{4}$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{\frac{3 \cdot 4}{4 \cdot 4} - \frac{9}{16}}$

Step 11.2.6.2

Multiply $4$ by $4$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{\frac{3 \cdot 4}{16} - \frac{9}{16}}$

Step 11.2.7

Combine the numerators over the common denominator.

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{\frac{3 \cdot 4 - 9}{16}}$

Step 11.2.8

Simplify the numerator.

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Step 11.2.8.1

Multiply $3$ by $4$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{\frac{12 - 9}{16}}$

Step 11.2.8.2

Subtract $9$ from $12$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \sqrt{\frac{3}{16}}$

Step 11.2.9

Rewrite $\sqrt{\frac{3}{16}}$ as $\frac{\sqrt{3}}{\sqrt{16}}$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \frac{\sqrt{3}}{\sqrt{16}}$

Step 11.2.10

Simplify the denominator.

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Step 11.2.10.1

Rewrite $16$ as $4^{2}$ .

$f (\frac{3}{4}) = \frac{9}{4} \cdot \frac{\sqrt{3}}{\sqrt{4^{2}}}$

Step 11.2.10.2

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{3}{4}) = \frac{9}{4} \cdot \frac{\sqrt{3}}{4}$

Step 11.2.11

Multiply $\frac{9}{4} \cdot \frac{\sqrt{3}}{4}$ .

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Step 11.2.11.1

Multiply $\frac{9}{4}$ by $\frac{\sqrt{3}}{4}$ .

$f (\frac{3}{4}) = \frac{9 \sqrt{3}}{4 \cdot 4}$

Step 11.2.11.2

Multiply $4$ by $4$ .

$f (\frac{3}{4}) = \frac{9 \sqrt{3}}{16}$

Step 11.2.12

The final answer is $\frac{9 \sqrt{3}}{16}$ .

$y = \frac{9 \sqrt{3}}{16}$

Step 12

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 {(- {(0)}^{2} + 0)}^{\frac{1}{2}} (1 - (0))}$

Step 13

Evaluate the second derivative.

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Step 13.1

Remove parentheses.

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 {(- {(0)}^{2} + 0)}^{\frac{1}{2}} (1 - (0))}$

Step 13.2

Simplify each term.

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Step 13.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 {(- 0 + 0)}^{\frac{1}{2}} (1 - (0))}$

Step 13.2.2

Multiply $- 1$ by $0$ .

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 {(0 + 0)}^{\frac{1}{2}} (1 - (0))}$

Step 13.3

Reduce the expression by cancelling the common factors.

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Step 13.3.1

Add $0$ and $0$ .

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 \cdot 0^{\frac{1}{2}} (1 - (0))}$

Step 13.3.2

Simplify the expression.

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Step 13.3.2.1

Rewrite $0$ as $0^{2}$ .

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 \cdot {(0^{2})}^{\frac{1}{2}} (1 - (0))}$

Step 13.3.2.2

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

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 \cdot 0^{2 (\frac{1}{2})} (1 - (0))}$

Step 13.3.3

Cancel the common factor of $2$ .

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Step 13.3.3.1

Cancel the common factor.

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 \cdot 0^{2 (\frac{1}{2})} (1 - (0))}$

Step 13.3.3.2

Rewrite the expression.

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 \cdot 0^{1} (1 - (0))}$

Step 13.3.4

Simplify the expression.

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Step 13.3.4.1

Evaluate the exponent.

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{4 \cdot 0 (1 - (0))}$

Step 13.3.4.2

Multiply $4$ by $0$ .

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{0 (1 - (0))}$

Step 13.3.4.3

Subtract $0$ from $1$ .

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{0 \cdot 1}$

Step 13.3.4.4

Multiply $0$ by $1$ .

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{0}$

Step 13.3.4.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{0}$

Step 13.3.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{3 (8 {(0)}^{2} - 12 \cdot 0 + 3)}{0}$

Step 13.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 14

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 15