Calculus Examples

$f (x) = x \sqrt{10 - x}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 - x}$ as ${(10 - x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [x {(10 - x)}^{\frac{1}{2}}]$

Step 1.1.2

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 $g (x) = {(10 - x)}^{\frac{1}{2}}$ .

$x \frac{d}{d x} [{(10 - x)}^{\frac{1}{2}}] + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.3

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 $g (x) = 10 - x$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $10 - x$ .

$x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.3.2

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

$x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.3.3

Replace all occurrences of $u$ with $10 - x$ .

$x (\frac{1}{2} {(10 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.4

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

$x (\frac{1}{2} {(10 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.5

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

$x (\frac{1}{2} {(10 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.6

Combine the numerators over the common denominator.

$x (\frac{1}{2} {(10 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

$x (\frac{1}{2} {(10 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.7.2

Subtract $2$ from $1$ .

$x (\frac{1}{2} {(10 - x)}^{\frac{- 1}{2}} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8

Combine fractions.

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Step 1.1.8.1

Move the negative in front of the fraction.

$x (\frac{1}{2} {(10 - x)}^{- \frac{1}{2}} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8.2

Combine $\frac{1}{2}$ and ${(10 - x)}^{- \frac{1}{2}}$ .

$x (\frac{{(10 - x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8.3

Move ${(10 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{1}{2 {(10 - x)}^{\frac{1}{2}}} \frac{d}{d x} [10 - x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.8.4

Combine $\frac{1}{2 {(10 - x)}^{\frac{1}{2}}}$ and $x$ .

$\frac{x}{2 {(10 - x)}^{\frac{1}{2}}} \frac{d}{d x} [10 - x] + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.9

By the Sum Rule, the derivative of $10 - x$ with respect to $x$ is $\frac{d}{d x} [10] + \frac{d}{d x} [- x]$ .

$\frac{x}{2 {(10 - x)}^{\frac{1}{2}}} (\frac{d}{d x} [10] + \frac{d}{d x} [- x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.10

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$\frac{x}{2 {(10 - x)}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.11

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{x}{2 {(10 - x)}^{\frac{1}{2}}} \frac{d}{d x} [- x] + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.12

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

$\frac{x}{2 {(10 - x)}^{\frac{1}{2}}} (- \frac{d}{d x} [x]) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.13

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

$\frac{x}{2 {(10 - x)}^{\frac{1}{2}}} (- 1 \cdot 1) + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14

Combine fractions.

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Step 1.1.14.1

Multiply $- 1$ by $1$ .

$\frac{x}{2 {(10 - x)}^{\frac{1}{2}}} \cdot - 1 + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14.2

Combine $\frac{x}{2 {(10 - x)}^{\frac{1}{2}}}$ and $- 1$ .

$\frac{x \cdot - 1}{2 {(10 - x)}^{\frac{1}{2}}} + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14.3

Simplify the expression.

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Step 1.1.14.3.1

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

$\frac{- 1 \cdot x}{2 {(10 - x)}^{\frac{1}{2}}} + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14.3.2

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

$\frac{- x}{2 {(10 - x)}^{\frac{1}{2}}} + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.14.3.3

Move the negative in front of the fraction.

$- \frac{x}{2 {(10 - x)}^{\frac{1}{2}}} + {(10 - x)}^{\frac{1}{2}} \frac{d}{d x} [x]$

Step 1.1.15

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

$- \frac{x}{2 {(10 - x)}^{\frac{1}{2}}} + {(10 - x)}^{\frac{1}{2}} \cdot 1$

Step 1.1.16

Multiply ${(10 - x)}^{\frac{1}{2}}$ by $1$ .

$- \frac{x}{2 {(10 - x)}^{\frac{1}{2}}} + {(10 - x)}^{\frac{1}{2}}$

Step 1.1.17

To write ${(10 - x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(10 - x)}^{\frac{1}{2}}}{2 {(10 - x)}^{\frac{1}{2}}}$ .

$- \frac{x}{2 {(10 - x)}^{\frac{1}{2}}} + {(10 - x)}^{\frac{1}{2}} \cdot \frac{2 {(10 - x)}^{\frac{1}{2}}}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.18

Combine ${(10 - x)}^{\frac{1}{2}}$ and $\frac{2 {(10 - x)}^{\frac{1}{2}}}{2 {(10 - x)}^{\frac{1}{2}}}$ .

$- \frac{x}{2 {(10 - x)}^{\frac{1}{2}}} + \frac{{(10 - x)}^{\frac{1}{2}} (2 {(10 - x)}^{\frac{1}{2}})}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.19

Combine the numerators over the common denominator.

$\frac{- x + {(10 - x)}^{\frac{1}{2}} (2 {(10 - x)}^{\frac{1}{2}})}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.20

Multiply ${(10 - x)}^{\frac{1}{2}}$ by ${(10 - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.1.20.1

Move ${(10 - x)}^{\frac{1}{2}}$ .

$\frac{- x + {(10 - x)}^{\frac{1}{2}} {(10 - x)}^{\frac{1}{2}} \cdot 2}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.20.2

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

$\frac{- x + {(10 - x)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.20.3

Combine the numerators over the common denominator.

$\frac{- x + {(10 - x)}^{\frac{1 + 1}{2}} \cdot 2}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.20.4

Add $1$ and $1$ .

$\frac{- x + {(10 - x)}^{\frac{2}{2}} \cdot 2}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.20.5

Divide $2$ by $2$ .

$\frac{- x + {(10 - x)}^{1} \cdot 2}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.21

Simplify ${(10 - x)}^{1} \cdot 2$ .

$\frac{- x + (10 - x) \cdot 2}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.22

Move $2$ to the left of $10 - x$ .

$\frac{- x + 2 \cdot (10 - x)}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.23

Simplify.

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Step 1.1.23.1

Apply the distributive property.

$\frac{- x + 2 \cdot 10 + 2 (- x)}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.23.2

Simplify the numerator.

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Step 1.1.23.2.1

Simplify each term.

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Step 1.1.23.2.1.1

Multiply $2$ by $10$ .

$\frac{- x + 20 + 2 (- x)}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.23.2.1.2

Multiply $- 1$ by $2$ .

$\frac{- x + 20 - 2 x}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.23.2.2

Subtract $2 x$ from $- x$ .

$\frac{- 3 x + 20}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.23.3

Factor $- 1$ out of $- 3 x$ .

$\frac{- (3 x) + 20}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.23.4

Rewrite $20$ as $- 1 (- 20)$ .

$\frac{- (3 x) - 1 (- 20)}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.23.5

Factor $- 1$ out of $- (3 x) - 1 (- 20)$ .

$\frac{- (3 x - 20)}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.23.6

Rewrite $- (3 x - 20)$ as $- 1 (3 x - 20)$ .

$\frac{- 1 (3 x - 20)}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.1.23.7

Move the negative in front of the fraction.

$f' (x) = - \frac{3 x - 20}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{3 x - 20}{2 {(10 - x)}^{\frac{1}{2}}}$ .

$- \frac{3 x - 20}{2 {(10 - x)}^{\frac{1}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{3 x - 20}{2 {(10 - x)}^{\frac{1}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{3 x - 20}{2 {(10 - x)}^{\frac{1}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$3 x - 20 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Add $20$ to both sides of the equation.

$3 x = 20$

Step 2.3.2

Divide each term in $3 x = 20$ by $3$ and simplify.

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Step 2.3.2.1

Divide each term in $3 x = 20$ by $3$ .

$\frac{3 x}{3} = \frac{20}{3}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{20}{3}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{20}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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

$- \frac{3 x - 20}{2 \sqrt{{(10 - x)}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$- \frac{3 x - 20}{2 \sqrt{10 - x}}$

Step 3.2

Set the denominator in $\frac{3 x - 20}{2 \sqrt{10 - x}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{10 - x} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

${(2 \sqrt{10 - x})}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 - x}$ as ${(10 - x)}^{\frac{1}{2}}$ .

${(2 {(10 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(2 {(10 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $2 {(10 - x)}^{\frac{1}{2}}$ .

$2^{2} {({(10 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2.1.2

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

$4 {({(10 - x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(10 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.2.1.3.1

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

$4 {(10 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$4 {(10 - x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$4 {(10 - x)}^{1} = 0^{2}$

Step 3.3.2.2.1.4

Simplify.

$4 (10 - x) = 0^{2}$

Step 3.3.2.2.1.5

Apply the distributive property.

$4 \cdot 10 + 4 (- x) = 0^{2}$

Step 3.3.2.2.1.6

Multiply.

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Step 3.3.2.2.1.6.1

Multiply $4$ by $10$ .

$40 + 4 (- x) = 0^{2}$

Step 3.3.2.2.1.6.2

Multiply $- 1$ by $4$ .

$40 - 4 x = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$40 - 4 x = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Subtract $40$ from both sides of the equation.

$- 4 x = - 40$

Step 3.3.3.2

Divide each term in $- 4 x = - 40$ by $- 4$ and simplify.

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Step 3.3.3.2.1

Divide each term in $- 4 x = - 40$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 40}{- 4}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 40}{- 4}$

Step 3.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 40}{- 4}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Divide $- 40$ by $- 4$ .

$x = 10$

Step 3.4

Set the radicand in $\sqrt{10 - x}$ less than $0$ to find where the expression is undefined.

$10 - x < 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

Subtract $10$ from both sides of the inequality.

$- x < - 10$

Step 3.5.2

Divide each term in $- x < - 10$ by $- 1$ and simplify.

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Step 3.5.2.1

Divide each term in $- x < - 10$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- 10}{- 1}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{- 10}{- 1}$

Step 3.5.2.2.2

Divide $x$ by $1$ .

$x > \frac{- 10}{- 1}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Divide $- 10$ by $- 1$ .

$x > 10$

Step 3.6

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 $0$ .

$x \geq 10$

$[10, \infty)$

$x \geq 10$

$[10, \infty)$

Step 4

Evaluate $x \sqrt{10 - x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{20}{3}$ .

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Step 4.1.1

Substitute $\frac{20}{3}$ for $x$ .

$(\frac{20}{3}) \sqrt{10 - (\frac{20}{3})}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{20}{3} \sqrt{10 \cdot \frac{3}{3} - \frac{20}{3}}$

Step 4.1.2.2

Combine $10$ and $\frac{3}{3}$ .

$\frac{20}{3} \sqrt{\frac{10 \cdot 3}{3} - \frac{20}{3}}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$\frac{20}{3} \sqrt{\frac{10 \cdot 3 - 20}{3}}$

Step 4.1.2.4

Simplify the numerator.

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Step 4.1.2.4.1

Multiply $10$ by $3$ .

$\frac{20}{3} \sqrt{\frac{30 - 20}{3}}$

Step 4.1.2.4.2

Subtract $20$ from $30$ .

$\frac{20}{3} \sqrt{\frac{10}{3}}$

Step 4.1.2.5

Rewrite $\sqrt{\frac{10}{3}}$ as $\frac{\sqrt{10}}{\sqrt{3}}$ .

$\frac{20}{3} \cdot \frac{\sqrt{10}}{\sqrt{3}}$

Step 4.1.2.6

Multiply $\frac{\sqrt{10}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{20}{3} (\frac{\sqrt{10}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 4.1.2.7

Combine and simplify the denominator.

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Step 4.1.2.7.1

Multiply $\frac{\sqrt{10}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.1.2.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.1.2.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.1.2.7.4

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

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.1.2.7.5

Add $1$ and $1$ .

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.1.2.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.1.2.7.6.1

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

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.1.2.7.6.2

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

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.1.2.7.6.3

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

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.1.2.7.6.4

Cancel the common factor of $2$ .

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Step 4.1.2.7.6.4.1

Cancel the common factor.

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.1.2.7.6.4.2

Rewrite the expression.

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{3^{1}}$

Step 4.1.2.7.6.5

Evaluate the exponent.

$\frac{20}{3} \cdot \frac{\sqrt{10} \sqrt{3}}{3}$

Step 4.1.2.8

Simplify the numerator.

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Step 4.1.2.8.1

Combine using the product rule for radicals.

$\frac{20}{3} \cdot \frac{\sqrt{10 \cdot 3}}{3}$

Step 4.1.2.8.2

Multiply $10$ by $3$ .

$\frac{20}{3} \cdot \frac{\sqrt{30}}{3}$

Step 4.1.2.9

Multiply $\frac{20}{3} \cdot \frac{\sqrt{30}}{3}$ .

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Step 4.1.2.9.1

Multiply $\frac{20}{3}$ by $\frac{\sqrt{30}}{3}$ .

$\frac{20 \sqrt{30}}{3 \cdot 3}$

Step 4.1.2.9.2

Multiply $3$ by $3$ .

$\frac{20 \sqrt{30}}{9}$

Step 4.2

Evaluate at $x = 10$ .

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Step 4.2.1

Substitute $10$ for $x$ .

$(10) \sqrt{10 - (10)}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $- 1$ by $10$ .

$10 \sqrt{10 - 10}$

Step 4.2.2.2

Subtract $10$ from $10$ .

$10 \sqrt{0}$

Step 4.2.2.3

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

$10 \sqrt{0^{2}}$

Step 4.2.2.4

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

$10 \cdot 0$

Step 4.2.2.5

Multiply $10$ by $0$ .

$0$

Step 4.3

List all of the points.

$(\frac{20}{3}, \frac{20 \sqrt{30}}{9}), (10, 0)$

Step 5