Calculus Examples

$x^{3} y^{2} = x y^{3} + 6$

Step 1

Set each solution of $x^{3} y^{2}$ as a function of $x$ .

$y = x y^{3} + 6 \to f (x) = x y^{3} + 6$

Step 2

Because the $y$ variable in the equation $x^{3} y^{2} = x y^{3} + 6$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 2.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} y^{2}) = \frac{d}{d x} (x y^{3} + 6)$

Step 2.2

Differentiate the left side of the equation.

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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 $f (x) = x^{3}$ and $g (x) = y^{2}$ .

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

Step 2.2.2

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

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Step 2.2.2.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

Move $2$ to the left of $x^{3}$ .

$2 \cdot x^{3} y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x^{3}]$

Step 2.2.4

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$2 x^{3} y y' + y^{2} \frac{d}{d x} [x^{3}]$

Step 2.2.5

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

$2 x^{3} y y' + y^{2} (3 x^{2})$

Step 2.2.6

Reorder.

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Step 2.2.6.1

Move $3$ to the left of $y^{2}$ .

$2 x^{3} y y' + 3 \cdot y^{2} x^{2}$

Step 2.2.6.2

Reorder terms.

$2 x^{3} y y' + 3 x^{2} y^{2}$

Step 2.3

Differentiate the right side of the equation.

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Step 2.3.1

By the Sum Rule, the derivative of $x y^{3} + 6$ with respect to $x$ is $\frac{d}{d x} [x y^{3}] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [x y^{3}] + \frac{d}{d x} [6]$

Step 2.3.2

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

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Step 2.3.2.1

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 $f (x) = x$ and $g (x) = y^{3}$ .

$x \frac{d}{d x} [y^{3}] + y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 2.3.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.3.2.2.1

To apply the Chain Rule, set $u$ as $y$ .

$x (\frac{d}{d u} [u^{3}] \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 2.3.2.2.2

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

$x (3 u^{2} \frac{d}{d x} [y]) + y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 2.3.2.2.3

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

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

Step 2.3.2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$x (3 y^{2} y') + y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 2.3.2.4

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

$x (3 y^{2} y') + y^{3} \cdot 1 + \frac{d}{d x} [6]$

Step 2.3.2.5

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

$3 \cdot x y^{2} y' + y^{3} \cdot 1 + \frac{d}{d x} [6]$

Step 2.3.2.6

Multiply $y^{3}$ by $1$ .

$3 x y^{2} y' + y^{3} + \frac{d}{d x} [6]$

Step 2.3.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$3 x y^{2} y' + y^{3} + 0$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Add $3 x y^{2} y' + y^{3}$ and $0$ .

$3 x y^{2} y' + y^{3}$

Step 2.3.4.2

Reorder terms.

$y^{3} + 3 y^{2} x y'$

Step 2.4

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

$2 x^{3} y y' + 3 x^{2} y^{2} = y^{3} + 3 y^{2} x y'$

Step 2.5

Solve for $y'$ .

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Step 2.5.1

Subtract $3 y^{2} x y'$ from both sides of the equation.

$2 x^{3} y y' + 3 x^{2} y^{2} - 3 y^{2} x y' = y^{3}$

Step 2.5.2

Subtract $3 x^{2} y^{2}$ from both sides of the equation.

$2 x^{3} y y' - 3 y^{2} x y' = y^{3} - 3 x^{2} y^{2}$

Step 2.5.3

Factor $x y y'$ out of $2 x^{3} y y' - 3 y^{2} x y'$ .

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Step 2.5.3.1

Factor $x y y'$ out of $2 x^{3} y y'$ .

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

Step 2.5.3.2

Factor $x y y'$ out of $- 3 y^{2} x y'$ .

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

Step 2.5.3.3

Factor $x y y'$ out of $x y y' (2 x^{2}) + x y y' (- 3 y)$ .

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

Step 2.5.4

Divide each term in $x y y' (2 x^{2} - 3 y) = y^{3} - 3 x^{2} y^{2}$ by $x y (2 x^{2} - 3 y)$ and simplify.

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Step 2.5.4.1

Divide each term in $x y y' (2 x^{2} - 3 y) = y^{3} - 3 x^{2} y^{2}$ by $x y (2 x^{2} - 3 y)$ .

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

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $x$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

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

Step 2.5.4.2.1.2

Rewrite the expression.

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

Step 2.5.4.2.2

Cancel the common factor of $y$ .

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Step 2.5.4.2.2.1

Cancel the common factor.

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

Step 2.5.4.2.2.2

Rewrite the expression.

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

Step 2.5.4.2.3

Cancel the common factor of $2 x^{2} - 3 y$ .

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Step 2.5.4.2.3.1

Cancel the common factor.

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

Step 2.5.4.2.3.2

Divide $y'$ by $1$ .

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

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Simplify each term.

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Step 2.5.4.3.1.1

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

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Step 2.5.4.3.1.1.1

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

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

Step 2.5.4.3.1.1.2

Cancel the common factors.

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Step 2.5.4.3.1.1.2.1

Factor $y$ out of $x y (2 x^{2} - 3 y)$ .

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

Step 2.5.4.3.1.1.2.2

Cancel the common factor.

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

Step 2.5.4.3.1.1.2.3

Rewrite the expression.

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

Step 2.5.4.3.1.2

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

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Step 2.5.4.3.1.2.1

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

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

Step 2.5.4.3.1.2.2

Cancel the common factors.

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Step 2.5.4.3.1.2.2.1

Factor $x$ out of $x y (2 x^{2} - 3 y)$ .

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

Step 2.5.4.3.1.2.2.2

Cancel the common factor.

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

Step 2.5.4.3.1.2.2.3

Rewrite the expression.

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

Step 2.5.4.3.1.3

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

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Step 2.5.4.3.1.3.1

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

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

Step 2.5.4.3.1.3.2

Cancel the common factors.

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Step 2.5.4.3.1.3.2.1

Cancel the common factor.

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

Step 2.5.4.3.1.3.2.2

Rewrite the expression.

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

Step 2.5.4.3.1.4

Move the negative in front of the fraction.

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

Step 2.5.4.3.2

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

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

Step 2.5.4.3.3

Write each expression with a common denominator of $x (2 x^{2} - 3 y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.4.3.3.1

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

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

Step 2.5.4.3.3.2

Reorder the factors of $(2 x^{2} - 3 y) x$ .

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

Step 2.5.4.3.4

Combine the numerators over the common denominator.

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

Step 2.5.4.3.5

Simplify the numerator.

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Step 2.5.4.3.5.1

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

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Step 2.5.4.3.5.1.1

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

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

Step 2.5.4.3.5.1.2

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

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

Step 2.5.4.3.5.1.3

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

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

Step 2.5.4.3.5.2

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

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Step 2.5.4.3.5.2.1

Move $x$ .

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

Step 2.5.4.3.5.2.2

Multiply $x$ by $x$ .

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

Step 2.6

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

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

Step 3

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

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Step 3.1

Set the numerator equal to zero.

$y (y - 3 x^{2}) = 0$

Step 3.2

Solve the equation for $x$ .

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Step 3.2.1

Divide each term in $y (y - 3 x^{2}) = 0$ by $y$ and simplify.

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Step 3.2.1.1

Divide each term in $y (y - 3 x^{2}) = 0$ by $y$ .

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

Step 3.2.1.2

Simplify the left side.

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Step 3.2.1.2.1

Cancel the common factor of $y$ .

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Step 3.2.1.2.1.1

Cancel the common factor.

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

Step 3.2.1.2.1.2

Divide $y - 3 x^{2}$ by $1$ .

$y - 3 x^{2} = \frac{0}{y}$

Step 3.2.1.3

Simplify the right side.

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Step 3.2.1.3.1

Divide $0$ by $y$ .

$y - 3 x^{2} = 0$

Step 3.2.2

Subtract $y$ from both sides of the equation.

$- 3 x^{2} = - y$

Step 3.2.3

Divide each term in $- 3 x^{2} = - y$ by $- 3$ and simplify.

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Step 3.2.3.1

Divide each term in $- 3 x^{2} = - y$ by $- 3$ .

$\frac{- 3 x^{2}}{- 3} = \frac{- y}{- 3}$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

$\frac{- 3 x^{2}}{- 3} = \frac{- y}{- 3}$

Step 3.2.3.2.1.2

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

$x^{2} = \frac{- y}{- 3}$

Step 3.2.3.3

Simplify the right side.

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Step 3.2.3.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{y}{3}$

Step 3.2.4

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

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

Step 3.2.5

Simplify $\pm \sqrt{\frac{y}{3}}$ .

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Step 3.2.5.1

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

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

Step 3.2.5.2

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

$x = \pm \frac{\sqrt{y}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.2.5.3

Combine and simplify the denominator.

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Step 3.2.5.3.1

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

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

Step 3.2.5.3.2

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

$x = \pm \frac{\sqrt{y} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.2.5.3.3

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

$x = \pm \frac{\sqrt{y} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.2.5.3.4

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

$x = \pm \frac{\sqrt{y} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.2.5.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{y} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.2.5.3.6

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

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Step 3.2.5.3.6.1

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

$x = \pm \frac{\sqrt{y} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.2.5.3.6.2

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

$x = \pm \frac{\sqrt{y} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.2.5.3.6.3

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

$x = \pm \frac{\sqrt{y} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.2.5.3.6.4

Cancel the common factor of $2$ .

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Step 3.2.5.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{y} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.2.5.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{y} \sqrt{3}}{3^{1}}$

Step 3.2.5.3.6.5

Evaluate the exponent.

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

Step 3.2.5.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{y \cdot 3}}{3}$

Step 3.2.5.5

Reorder factors in $\pm \frac{\sqrt{y \cdot 3}}{3}$ .

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

Step 3.2.6

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

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Step 3.2.6.1

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

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

Step 3.2.6.2

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

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

Step 3.2.6.3

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

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

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

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

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

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

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

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

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

Step 4

Solve the function $f (x) = x y^{3} + 6$ at $x = \frac{\sqrt{3 y}}{3}$ .

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Step 4.1

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

$f (\frac{\sqrt{3 y}}{3}) = (\frac{\sqrt{3 y}}{3}) \cdot y^{3} + 6$

Step 4.2

Simplify the result.

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Step 4.2.1

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

$f (\frac{\sqrt{3 y}}{3}) = \frac{\sqrt{3 y} y^{3}}{3} + 6$

Step 4.2.2

The final answer is $\frac{\sqrt{3 y} y^{3}}{3} + 6$ .

$\frac{\sqrt{3 y} y^{3}}{3} + 6$

Step 5

Solve the function $f (x) = x y^{3} + 6$ at $x = - \frac{\sqrt{3 y}}{3}$ .

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Step 5.1

Replace the variable $x$ with $- \frac{\sqrt{3 y}}{3}$ in the expression.

$f (- \frac{\sqrt{3 y}}{3}) = (- \frac{\sqrt{3 y}}{3}) \cdot y^{3} + 6$

Step 5.2

Simplify the result.

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Step 5.2.1

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

$f (- \frac{\sqrt{3 y}}{3}) = - \frac{y^{3} \sqrt{3 y}}{3} + 6$

Step 5.2.2

The final answer is $- \frac{y^{3} \sqrt{3 y}}{3} + 6$ .

$- \frac{y^{3} \sqrt{3 y}}{3} + 6$

Step 6

The horizontal tangent lines are $y = \frac{\sqrt{3 y} y^{3}}{3} + 6, y = - \frac{y^{3} \sqrt{3 y}}{3} + 6$

$y = \frac{\sqrt{3 y} y^{3}}{3} + 6, y = - \frac{y^{3} \sqrt{3 y}}{3} + 6$

Step 7