Calculus Examples

$x = \frac{1}{3} \sqrt{y} (y - 3)$ , $1 \leq y \leq 9$

Step 1

Remove parentheses.

$x = \frac{1}{3} \cdot (\sqrt{y} (y - 3))$

Step 2

Remove parentheses.

$x = \frac{1}{3} \cdot (\sqrt{y} (y - 3))$

Step 3

Simplify $\frac{1}{3} \cdot (\sqrt{y} (y - 3))$ .

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Step 3.1

Apply the distributive property.

$x = \frac{1}{3} \cdot (\sqrt{y} y + \sqrt{y} \cdot - 3)$

Step 3.2

Move $- 3$ to the left of $\sqrt{y}$ .

$x = \frac{1}{3} \cdot (\sqrt{y} y - 3 \cdot \sqrt{y})$

Step 3.3

Apply the distributive property.

$x = \frac{1}{3} (\sqrt{y} y) + \frac{1}{3} (- 3 \sqrt{y})$

Step 3.4

Multiply $\frac{1}{3} (\sqrt{y} y)$ .

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Step 3.4.1

Combine $\sqrt{y}$ and $\frac{1}{3}$ .

$x = \frac{\sqrt{y}}{3} y + \frac{1}{3} (- 3 \sqrt{y})$

Step 3.4.2

Combine $\frac{\sqrt{y}}{3}$ and $y$ .

$x = \frac{\sqrt{y} y}{3} + \frac{1}{3} (- 3 \sqrt{y})$

Step 3.5

Cancel the common factor of $3$ .

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Step 3.5.1

Factor $3$ out of $- 3 \sqrt{y}$ .

$x = \frac{\sqrt{y} y}{3} + \frac{1}{3} (3 (- \sqrt{y}))$

Step 3.5.2

Cancel the common factor.

$x = \frac{\sqrt{y} y}{3} + \frac{1}{3} (3 (- \sqrt{y}))$

Step 3.5.3

Rewrite the expression.

$x = \frac{\sqrt{y} y}{3} - \sqrt{y}$

Step 4

Check if $f (y) = \frac{\sqrt{y} y}{3} - \sqrt{y}$ is continuous.

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Step 4.1

To find whether the function is continuous on $[1, 9]$ or not, find the domain of $f (y) = \frac{\sqrt{y} y}{3} - \sqrt{y}$ .

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Step 4.1.1

Set the radicand in $\sqrt{y}$ greater than or equal to $0$ to find where the expression is defined.

$y \geq 0$

Step 4.1.2

The domain is all values of $y$ that make the expression defined.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Step 4.2

$f (y)$ is continuous on $[1, 9]$ .

The function is continuous.

Step 5

Check if $f (y) = \frac{\sqrt{y} y}{3} - \sqrt{y}$ is differentiable.

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Step 5.1

Find the derivative.

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Step 5.1.1

Find the first derivative.

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Step 5.1.1.1

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

$\frac{d}{d y} [\frac{\sqrt{y} y}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2

Evaluate $\frac{d}{d y} [\frac{\sqrt{y} y}{3}]$ .

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Step 5.1.1.2.1

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

$\frac{d}{d y} [\frac{y^{\frac{1}{2}} y}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.2

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

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Step 5.1.1.2.2.1

Multiply $y^{\frac{1}{2}}$ by $y$ .

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Step 5.1.1.2.2.1.1

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

$\frac{d}{d y} [\frac{y^{\frac{1}{2}} y^{1}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.2.1.2

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

$\frac{d}{d y} [\frac{y^{\frac{1}{2} + 1}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.2.2

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

$\frac{d}{d y} [\frac{y^{\frac{1}{2} + \frac{2}{2}}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.2.3

Combine the numerators over the common denominator.

$\frac{d}{d y} [\frac{y^{\frac{1 + 2}{2}}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.2.4

Add $1$ and $2$ .

$\frac{d}{d y} [\frac{y^{\frac{3}{2}}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.3

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

$\frac{1}{3} \frac{d}{d y} [y^{\frac{3}{2}}] + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.4

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

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

Step 5.1.1.2.5

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

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

Step 5.1.1.2.6

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

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

Step 5.1.1.2.7

Combine the numerators over the common denominator.

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

Step 5.1.1.2.8

Simplify the numerator.

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Step 5.1.1.2.8.1

Multiply $- 1$ by $2$ .

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

Step 5.1.1.2.8.2

Subtract $2$ from $3$ .

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

Step 5.1.1.2.9

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

$\frac{1}{3} \cdot \frac{3 y^{\frac{1}{2}}}{2} + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.10

Multiply $\frac{1}{3}$ by $\frac{3 y^{\frac{1}{2}}}{2}$ .

$\frac{3 y^{\frac{1}{2}}}{3 \cdot 2} + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.11

Multiply $3$ by $2$ .

$\frac{3 y^{\frac{1}{2}}}{6} + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.12

Factor $3$ out of $3 y^{\frac{1}{2}}$ .

$\frac{3 (y^{\frac{1}{2}})}{6} + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.13

Cancel the common factors.

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Step 5.1.1.2.13.1

Factor $3$ out of $6$ .

$\frac{3 y^{\frac{1}{2}}}{3 \cdot 2} + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.13.2

Cancel the common factor.

$\frac{3 y^{\frac{1}{2}}}{3 \cdot 2} + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.2.13.3

Rewrite the expression.

$\frac{y^{\frac{1}{2}}}{2} + \frac{d}{d y} [- \sqrt{y}]$

Step 5.1.1.3

Evaluate $\frac{d}{d y} [- \sqrt{y}]$ .

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Step 5.1.1.3.1

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

$\frac{y^{\frac{1}{2}}}{2} + \frac{d}{d y} [- y^{\frac{1}{2}}]$

Step 5.1.1.3.2

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

$\frac{y^{\frac{1}{2}}}{2} - \frac{d}{d y} [y^{\frac{1}{2}}]$

Step 5.1.1.3.3

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

$\frac{y^{\frac{1}{2}}}{2} - (\frac{1}{2} y^{\frac{1}{2} - 1})$

Step 5.1.1.3.4

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

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

Step 5.1.1.3.5

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

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

Step 5.1.1.3.6

Combine the numerators over the common denominator.

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

Step 5.1.1.3.7

Simplify the numerator.

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Step 5.1.1.3.7.1

Multiply $- 1$ by $2$ .

$\frac{y^{\frac{1}{2}}}{2} - (\frac{1}{2} y^{\frac{1 - 2}{2}})$

Step 5.1.1.3.7.2

Subtract $2$ from $1$ .

$\frac{y^{\frac{1}{2}}}{2} - (\frac{1}{2} y^{\frac{- 1}{2}})$

Step 5.1.1.3.8

Move the negative in front of the fraction.

$\frac{y^{\frac{1}{2}}}{2} - (\frac{1}{2} y^{- \frac{1}{2}})$

Step 5.1.1.3.9

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

$\frac{y^{\frac{1}{2}}}{2} - \frac{y^{- \frac{1}{2}}}{2}$

Step 5.1.1.3.10

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

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

Step 5.1.2

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

$\frac{y^{\frac{1}{2}}}{2} - \frac{1}{2 y^{\frac{1}{2}}}$

Step 5.2

Find if the derivative is continuous on $[1, 9]$ .

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Step 5.2.1

To find whether the function is continuous on $[1, 9]$ or not, find the domain of $f' (y) = \frac{y^{\frac{1}{2}}}{2} - \frac{1}{2 y^{\frac{1}{2}}}$ .

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Step 5.2.1.1

Convert expressions with fractional exponents to radicals.

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Step 5.2.1.1.1

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

$\frac{\sqrt{y^{1}}}{2} - \frac{1}{2 y^{\frac{1}{2}}}$

Step 5.2.1.1.2

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

$\frac{\sqrt{y^{1}}}{2} - \frac{1}{2 \sqrt{y^{1}}}$

Step 5.2.1.1.3

Anything raised to $1$ is the base itself.

$\frac{\sqrt{y}}{2} - \frac{1}{2 \sqrt{y^{1}}}$

Step 5.2.1.1.4

Anything raised to $1$ is the base itself.

$\frac{\sqrt{y}}{2} - \frac{1}{2 \sqrt{y}}$

Step 5.2.1.2

Set the radicand in $\sqrt{y}$ greater than or equal to $0$ to find where the expression is defined.

$y \geq 0$

Step 5.2.1.3

Set the denominator in $\frac{1}{2 \sqrt{y}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{y} = 0$

Step 5.2.1.4

Solve for $y$ .

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Step 5.2.1.4.1

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

${(2 \sqrt{y})}^{2} = 0^{2}$

Step 5.2.1.4.2

Simplify each side of the equation.

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Step 5.2.1.4.2.1

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

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

Step 5.2.1.4.2.2

Simplify the left side.

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Step 5.2.1.4.2.2.1

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

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Step 5.2.1.4.2.2.1.1

Apply the product rule to $2 y^{\frac{1}{2}}$ .

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

Step 5.2.1.4.2.2.1.2

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

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

Step 5.2.1.4.2.2.1.3

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

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Step 5.2.1.4.2.2.1.3.1

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

$4 y^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 5.2.1.4.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.4.2.2.1.3.2.1

Cancel the common factor.

$4 y^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 5.2.1.4.2.2.1.3.2.2

Rewrite the expression.

$4 y^{1} = 0^{2}$

Step 5.2.1.4.2.2.1.4

Simplify.

$4 y = 0^{2}$

Step 5.2.1.4.2.3

Simplify the right side.

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Step 5.2.1.4.2.3.1

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

$4 y = 0$

Step 5.2.1.4.3

Divide each term in $4 y = 0$ by $4$ and simplify.

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Step 5.2.1.4.3.1

Divide each term in $4 y = 0$ by $4$ .

$\frac{4 y}{4} = \frac{0}{4}$

Step 5.2.1.4.3.2

Simplify the left side.

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Step 5.2.1.4.3.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.4.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{0}{4}$

Step 5.2.1.4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{4}$

Step 5.2.1.4.3.3

Simplify the right side.

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Step 5.2.1.4.3.3.1

Divide $0$ by $4$ .

$y = 0$

Step 5.2.1.5

The domain is all values of $y$ that make the expression defined.

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${y | y > 0}$

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${y | y > 0}$

Step 5.2.2

$f' (y)$ is continuous on $[1, 9]$ .

The function is continuous.

Step 5.3

The function is differentiable on $[1, 9]$ because the derivative is continuous on $[1, 9]$ .

The function is differentiable.

Step 6

For arc length to be guaranteed, the function and its derivative must both be continuous on the closed interval $[1, 9]$ .

The function and its derivative are continuous on the closed interval $[1, 9]$ .

Step 7

Find the derivative of $f (y) = \frac{\sqrt{y} y}{3} - \sqrt{y}$ .

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Step 7.1

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

$\frac{d}{d y} [\frac{\sqrt{y} y}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2

Evaluate $\frac{d}{d y} [\frac{\sqrt{y} y}{3}]$ .

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Step 7.2.1

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

$\frac{d}{d y} [\frac{y^{\frac{1}{2}} y}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.2

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

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Step 7.2.2.1

Multiply $y^{\frac{1}{2}}$ by $y$ .

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Step 7.2.2.1.1

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

$\frac{d}{d y} [\frac{y^{\frac{1}{2}} y^{1}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.2.1.2

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

$\frac{d}{d y} [\frac{y^{\frac{1}{2} + 1}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.2.2

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

$\frac{d}{d y} [\frac{y^{\frac{1}{2} + \frac{2}{2}}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.2.3

Combine the numerators over the common denominator.

$\frac{d}{d y} [\frac{y^{\frac{1 + 2}{2}}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.2.4

Add $1$ and $2$ .

$\frac{d}{d y} [\frac{y^{\frac{3}{2}}}{3}] + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.3

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

$\frac{1}{3} \frac{d}{d y} [y^{\frac{3}{2}}] + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.4

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

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

Step 7.2.5

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

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

Step 7.2.6

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

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

Step 7.2.7

Combine the numerators over the common denominator.

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

Step 7.2.8

Simplify the numerator.

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Step 7.2.8.1

Multiply $- 1$ by $2$ .

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

Step 7.2.8.2

Subtract $2$ from $3$ .

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

Step 7.2.9

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

$\frac{1}{3} \cdot \frac{3 y^{\frac{1}{2}}}{2} + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.10

Multiply $\frac{1}{3}$ by $\frac{3 y^{\frac{1}{2}}}{2}$ .

$\frac{3 y^{\frac{1}{2}}}{3 \cdot 2} + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.11

Multiply $3$ by $2$ .

$\frac{3 y^{\frac{1}{2}}}{6} + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.12

Factor $3$ out of $3 y^{\frac{1}{2}}$ .

$\frac{3 (y^{\frac{1}{2}})}{6} + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.13

Cancel the common factors.

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Step 7.2.13.1

Factor $3$ out of $6$ .

$\frac{3 y^{\frac{1}{2}}}{3 \cdot 2} + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.13.2

Cancel the common factor.

$\frac{3 y^{\frac{1}{2}}}{3 \cdot 2} + \frac{d}{d y} [- \sqrt{y}]$

Step 7.2.13.3

Rewrite the expression.

$\frac{y^{\frac{1}{2}}}{2} + \frac{d}{d y} [- \sqrt{y}]$

Step 7.3

Evaluate $\frac{d}{d y} [- \sqrt{y}]$ .

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Step 7.3.1

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

$\frac{y^{\frac{1}{2}}}{2} + \frac{d}{d y} [- y^{\frac{1}{2}}]$

Step 7.3.2

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

$\frac{y^{\frac{1}{2}}}{2} - \frac{d}{d y} [y^{\frac{1}{2}}]$

Step 7.3.3

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

$\frac{y^{\frac{1}{2}}}{2} - (\frac{1}{2} y^{\frac{1}{2} - 1})$

Step 7.3.4

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

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

Step 7.3.5

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

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

Step 7.3.6

Combine the numerators over the common denominator.

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

Step 7.3.7

Simplify the numerator.

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Step 7.3.7.1

Multiply $- 1$ by $2$ .

$\frac{y^{\frac{1}{2}}}{2} - (\frac{1}{2} y^{\frac{1 - 2}{2}})$

Step 7.3.7.2

Subtract $2$ from $1$ .

$\frac{y^{\frac{1}{2}}}{2} - (\frac{1}{2} y^{\frac{- 1}{2}})$

Step 7.3.8

Move the negative in front of the fraction.

$\frac{y^{\frac{1}{2}}}{2} - (\frac{1}{2} y^{- \frac{1}{2}})$

Step 7.3.9

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

$\frac{y^{\frac{1}{2}}}{2} - \frac{y^{- \frac{1}{2}}}{2}$

Step 7.3.10

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

$\frac{y^{\frac{1}{2}}}{2} - \frac{1}{2 y^{\frac{1}{2}}}$

Step 8

To find the arc length of a function, use the formula $L = \int_{a}^{b} \sqrt{1 + {(f' (y))}^{2}} d y$ .

$\int_{1}^{9} \sqrt{1 + {(\frac{y^{\frac{1}{2}}}{2} - \frac{1}{2 y^{\frac{1}{2}}})}^{2}} d y$

Step 9