Calculus Examples

$f (x) = x^{- 1} + x y - 8 y^{- 1}$

Step 1

Find the first derivative of the function.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d x} [x^{- 1} + x y - 8 \frac{1}{y}]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{- 1} + x y - 8 \frac{1}{y}$ with respect to $x$ is $\frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [x y] + \frac{d}{d x} [- 8 \frac{1}{y}]$ .

$\frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [x y] + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 1.2.2

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

$- x^{- 2} + \frac{d}{d x} [x y] + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 1.3

Evaluate $\frac{d}{d x} [x y]$ .

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Step 1.3.1

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

$- x^{- 2} + y \frac{d}{d x} [x] + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 1.3.2

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

$- x^{- 2} + y \cdot 1 + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 1.3.3

Multiply $y$ by $1$ .

$- x^{- 2} + y + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 1.4

Evaluate $\frac{d}{d x} [- 8 \frac{1}{y}]$ .

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Step 1.4.1

Combine $- 8$ and $\frac{1}{y}$ .

$- x^{- 2} + y + \frac{d}{d x} [\frac{- 8}{y}]$

Step 1.4.2

Move the negative in front of the fraction.

$- x^{- 2} + y + \frac{d}{d x} [- \frac{8}{y}]$

Step 1.4.3

Since $- \frac{8}{y}$ is constant with respect to $x$ , the derivative of $- \frac{8}{y}$ with respect to $x$ is $0$ .

$- x^{- 2} + y + 0$

Step 1.5

Simplify.

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Step 1.5.1

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

$- x^{- 2} + y$

Step 1.5.2

Reorder terms.

$y - x^{- 2}$

Step 1.5.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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Step 2.2.6.1

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

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

Step 2.2.6.2

Multiply $2$ by $- 2$ .

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

Step 2.2.7

Multiply $2$ by $- 1$ .

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

Step 2.2.8

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

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

Step 2.2.9

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

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

Step 2.2.10

Subtract $4$ from $1$ .

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

Step 2.2.11

Multiply $- 2$ by $- 1$ .

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

Step 2.2.12

Multiply $\frac{1}{x^{2}}$ by $0$ .

$f'' (x) = 0 + 2 x^{- 3} + 0$

Step 2.2.13

Add $2 x^{- 3}$ and $0$ .

$f'' (x) = 0 + 2 x^{- 3}$

Step 2.3

Simplify.

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Step 2.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.3.2

Combine terms.

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Step 2.3.2.1

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

$f'' (x) = 0 + \frac{2}{x^{3}}$

Step 2.3.2.2

Add $0$ and $\frac{2}{x^{3}}$ .

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

Step 3

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

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

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d x} [x^{- 1} + x y - 8 \frac{1}{y}]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $x^{- 1} + x y - 8 \frac{1}{y}$ with respect to $x$ is $\frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [x y] + \frac{d}{d x} [- 8 \frac{1}{y}]$ .

$\frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [x y] + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 4.1.2.2

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

$- x^{- 2} + \frac{d}{d x} [x y] + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [x y]$ .

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Step 4.1.3.1

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

$- x^{- 2} + y \frac{d}{d x} [x] + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 4.1.3.2

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

$- x^{- 2} + y \cdot 1 + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 4.1.3.3

Multiply $y$ by $1$ .

$- x^{- 2} + y + \frac{d}{d x} [- 8 \frac{1}{y}]$

Step 4.1.4

Evaluate $\frac{d}{d x} [- 8 \frac{1}{y}]$ .

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Step 4.1.4.1

Combine $- 8$ and $\frac{1}{y}$ .

$- x^{- 2} + y + \frac{d}{d x} [\frac{- 8}{y}]$

Step 4.1.4.2

Move the negative in front of the fraction.

$- x^{- 2} + y + \frac{d}{d x} [- \frac{8}{y}]$

Step 4.1.4.3

Since $- \frac{8}{y}$ is constant with respect to $x$ , the derivative of $- \frac{8}{y}$ with respect to $x$ is $0$ .

$- x^{- 2} + y + 0$

Step 4.1.5

Simplify.

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Step 4.1.5.1

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

$- x^{- 2} + y$

Step 4.1.5.2

Reorder terms.

$y - x^{- 2}$

Step 4.1.5.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Subtract $y$ from both sides of the equation.

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

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2}, 1$

Step 5.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 5.4

Multiply each term in $- \frac{1}{x^{2}} = - y$ by $x^{2}$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $- \frac{1}{x^{2}} = - y$ by $x^{2}$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x^{2}$ .

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Step 5.4.2.1.1

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

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

Step 5.4.2.1.2

Cancel the common factor.

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

Step 5.4.2.1.3

Rewrite the expression.

$- 1 = - y x^{2}$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $- y x^{2} = - 1$ .

$- y x^{2} = - 1$

Step 5.5.2

Divide each term in $- y x^{2} = - 1$ by $- y$ and simplify.

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Step 5.5.2.1

Divide each term in $- y x^{2} = - 1$ by $- y$ .

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

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.5.2.2.2

Cancel the common factor of $y$ .

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Step 5.5.2.2.2.1

Cancel the common factor.

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

Step 5.5.2.2.2.2

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

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

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{1}{y}$

Step 5.5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{y}}$

Step 5.5.4

Simplify $\pm \sqrt{\frac{1}{y}}$ .

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Step 5.5.4.1

Rewrite $\sqrt{\frac{1}{y}}$ as $\frac{\sqrt{1}}{\sqrt{y}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{y}}$

Step 5.5.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{y}}$

Step 5.5.4.3

Multiply $\frac{1}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$x = \pm \frac{1}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}$

Step 5.5.4.4

Combine and simplify the denominator.

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Step 5.5.4.4.1

Multiply $\frac{1}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$x = \pm \frac{\sqrt{y}}{\sqrt{y} \sqrt{y}}$

Step 5.5.4.4.2

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

$x = \pm \frac{\sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}}$

Step 5.5.4.4.3

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

$x = \pm \frac{\sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}}$

Step 5.5.4.4.4

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

$x = \pm \frac{\sqrt{y}}{{\sqrt{y}}^{1 + 1}}$

Step 5.5.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{y}}{{\sqrt{y}}^{2}}$

Step 5.5.4.4.6

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

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Step 5.5.4.4.6.1

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

$x = \pm \frac{\sqrt{y}}{{(y^{\frac{1}{2}})}^{2}}$

Step 5.5.4.4.6.2

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

$x = \pm \frac{\sqrt{y}}{y^{\frac{1}{2} \cdot 2}}$

Step 5.5.4.4.6.3

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

$x = \pm \frac{\sqrt{y}}{y^{\frac{2}{2}}}$

Step 5.5.4.4.6.4

Cancel the common factor of $2$ .

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Step 5.5.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{y}}{y^{\frac{2}{2}}}$

Step 5.5.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{y}}{y^{1}}$

Step 5.5.4.4.6.5

Simplify.

$x = \pm \frac{\sqrt{y}}{y}$

Step 5.5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.5.5.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 5.5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 5.5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

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

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

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

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

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$x^{2} = 0$

Step 6.2

Solve for $y$ .

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Step 6.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

$x = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.3

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 $0$ .

$y = 0$

Step 7

Critical points to evaluate.

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

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

Step 8

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

$\frac{2}{{(\frac{\sqrt{y}}{y})}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

Apply the product rule to $\frac{\sqrt{y}}{y}$ .

$\frac{2}{\frac{{\sqrt{y}}^{3}}{y^{3}}}$

Step 9.1.2

Simplify the numerator.

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Step 9.1.2.1

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

$\frac{2}{\frac{\sqrt{y^{3}}}{y^{3}}}$

Step 9.1.2.2

Factor out $y^{2}$ .

$\frac{2}{\frac{\sqrt{y^{2} y}}{y^{3}}}$

Step 9.1.2.3

Pull terms out from under the radical.

$\frac{2}{\frac{y \sqrt{y}}{y^{3}}}$

Step 9.1.3

Cancel the common factor of $y$ and $y^{3}$ .

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Step 9.1.3.1

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

$\frac{2}{\frac{y (\sqrt{y})}{y^{3}}}$

Step 9.1.3.2

Cancel the common factors.

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Step 9.1.3.2.1

Factor $y$ out of $y^{3}$ .

$\frac{2}{\frac{y (\sqrt{y})}{y \cdot y^{2}}}$

Step 9.1.3.2.2

Cancel the common factor.

$\frac{2}{\frac{y \sqrt{y}}{y \cdot y^{2}}}$

Step 9.1.3.2.3

Rewrite the expression.

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

Step 9.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.3

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

$2 (\frac{y^{2}}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}})$

Step 9.4

Combine and simplify the denominator.

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Step 9.4.1

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

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

Step 9.4.2

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

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

Step 9.4.3

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

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

Step 9.4.4

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

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

Step 9.4.5

Add $1$ and $1$ .

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

Step 9.4.6

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

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Step 9.4.6.1

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

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

Step 9.4.6.2

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

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

Step 9.4.6.3

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

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

Step 9.4.6.4

Cancel the common factor of $2$ .

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Step 9.4.6.4.1

Cancel the common factor.

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

Step 9.4.6.4.2

Rewrite the expression.

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

Step 9.4.6.5

Simplify.

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

Step 9.5

Cancel the common factor of $y^{2}$ and $y$ .

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Step 9.5.1

Factor $y$ out of $y^{2} \sqrt{y}$ .

$2 \frac{y (y \sqrt{y})}{y}$

Step 9.5.2

Cancel the common factors.

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Step 9.5.2.1

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

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

Step 9.5.2.2

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

$2 \frac{y (y \sqrt{y})}{y \cdot 1}$

Step 9.5.2.3

Cancel the common factor.

$2 \frac{y (y \sqrt{y})}{y \cdot 1}$

Step 9.5.2.4

Rewrite the expression.

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

Step 9.5.2.5

Divide $y \sqrt{y}$ by $1$ .

$2 (y \sqrt{y})$

$2 y \sqrt{y}$

Step 10

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

No Local Extrema

Step 11