Calculus Examples

$7 x^{2} + 6 x y + 9 y^{2} = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (7 x^{2} + 6 x y + 9 y^{2}) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $7 x^{2} + 6 x y + 9 y^{2}$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}]$ .

$\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}]$

Step 2.2

Evaluate $\frac{d}{d x} [7 x^{2}]$ .

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Step 2.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}]$

Step 2.2.2

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

$7 (2 x) + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}]$

Step 2.2.3

Multiply $2$ by $7$ .

$14 x + \frac{d}{d x} [6 x y] + \frac{d}{d x} [9 y^{2}]$

Step 2.3

Evaluate $\frac{d}{d x} [6 x y]$ .

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Step 2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x y$ with respect to $x$ is $6 \frac{d}{d x} [x y]$ .

$14 x + 6 \frac{d}{d x} [x y] + \frac{d}{d x} [9 y^{2}]$

Step 2.3.2

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 $g (x) = y$ .

$14 x + 6 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [9 y^{2}]$

Step 2.3.3

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

$14 x + 6 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [9 y^{2}]$

Step 2.3.4

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

$14 x + 6 (x y' + y \cdot 1) + \frac{d}{d x} [9 y^{2}]$

Step 2.3.5

Multiply $y$ by $1$ .

$14 x + 6 (x y' + y) + \frac{d}{d x} [9 y^{2}]$

Step 2.4

Evaluate $\frac{d}{d x} [9 y^{2}]$ .

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Step 2.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9 y^{2}$ with respect to $x$ is $9 \frac{d}{d x} [y^{2}]$ .

$14 x + 6 (x y' + y) + 9 \frac{d}{d x} [y^{2}]$

Step 2.4.2

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 2.4.2.1

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

$14 x + 6 (x y' + y) + 9 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 2.4.2.2

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

$14 x + 6 (x y' + y) + 9 (2 u \frac{d}{d x} [y])$

Step 2.4.2.3

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

$14 x + 6 (x y' + y) + 9 (2 y \frac{d}{d x} [y])$

Step 2.4.3

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

$14 x + 6 (x y' + y) + 9 (2 y y')$

Step 2.4.4

Multiply $2$ by $9$ .

$14 x + 6 (x y' + y) + 18 y y'$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$14 x + 6 (x y') + 6 y + 18 y y'$

Step 2.5.2

Remove unnecessary parentheses.

$14 x + 6 x y' + 6 y + 18 y y'$

Step 2.5.3

Reorder terms.

$6 x y' + 18 y y' + 14 x + 6 y$

Step 3

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

$0$

Step 4

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

$6 x y' + 18 y y' + 14 x + 6 y = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.1.1

Subtract $14 x$ from both sides of the equation.

$6 x y' + 18 y y' + 6 y = - 14 x$

Step 5.1.2

Subtract $6 y$ from both sides of the equation.

$6 x y' + 18 y y' = - 14 x - 6 y$

Step 5.2

Factor $6 y'$ out of $6 x y' + 18 y y'$ .

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Step 5.2.1

Factor $6 y'$ out of $6 x y'$ .

$6 y' x + 18 y y' = - 14 x - 6 y$

Step 5.2.2

Factor $6 y'$ out of $18 y y'$ .

$6 y' x + 6 y' (3 y) = - 14 x - 6 y$

Step 5.2.3

Factor $6 y'$ out of $6 y' x + 6 y' (3 y)$ .

$6 y' (x + 3 y) = - 14 x - 6 y$

Step 5.3

Divide each term in $6 y' (x + 3 y) = - 14 x - 6 y$ by $6 (x + 3 y)$ and simplify.

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Step 5.3.1

Divide each term in $6 y' (x + 3 y) = - 14 x - 6 y$ by $6 (x + 3 y)$ .

$\frac{6 y' (x + 3 y)}{6 (x + 3 y)} = \frac{- 14 x}{6 (x + 3 y)} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $6$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{6 y' (x + 3 y)}{6 (x + 3 y)} = \frac{- 14 x}{6 (x + 3 y)} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (x + 3 y)}{x + 3 y} = \frac{- 14 x}{6 (x + 3 y)} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.2.2

Cancel the common factor of $x + 3 y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (x + 3 y)}{x + 3 y} = \frac{- 14 x}{6 (x + 3 y)} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 14 x}{6 (x + 3 y)} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $- 14$ and $6$ .

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Step 5.3.3.1.1.1

Factor $2$ out of $- 14 x$ .

$y' = \frac{2 (- 7 x)}{6 (x + 3 y)} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Factor $2$ out of $6 (x + 3 y)$ .

$y' = \frac{2 (- 7 x)}{2 (3 (x + 3 y))} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{2 (- 7 x)}{2 (3 (x + 3 y))} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- 7 x}{3 (x + 3 y)} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{7 x}{3 (x + 3 y)} + \frac{- 6 y}{6 (x + 3 y)}$

Step 5.3.3.1.3

Cancel the common factor of $- 6$ and $6$ .

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Step 5.3.3.1.3.1

Factor $6$ out of $- 6 y$ .

$y' = - \frac{7 x}{3 (x + 3 y)} + \frac{6 (- y)}{6 (x + 3 y)}$

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

Cancel the common factor.

$y' = - \frac{7 x}{3 (x + 3 y)} + \frac{6 (- y)}{6 (x + 3 y)}$

Step 5.3.3.1.3.2.2

Rewrite the expression.

$y' = - \frac{7 x}{3 (x + 3 y)} + \frac{- y}{x + 3 y}$

Step 5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{7 x}{3 (x + 3 y)} - \frac{y}{x + 3 y}$

Step 5.3.3.2

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

$y' = - \frac{7 x}{3 (x + 3 y)} - \frac{y}{x + 3 y} \cdot \frac{3}{3}$

Step 5.3.3.3

Write each expression with a common denominator of $3 (x + 3 y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.3.1

Multiply $\frac{y}{x + 3 y}$ by $\frac{3}{3}$ .

$y' = - \frac{7 x}{3 (x + 3 y)} - \frac{y \cdot 3}{(x + 3 y) \cdot 3}$

Step 5.3.3.3.2

Reorder the factors of $(x + 3 y) \cdot 3$ .

$y' = - \frac{7 x}{3 (x + 3 y)} - \frac{y \cdot 3}{3 (x + 3 y)}$

Step 5.3.3.4

Combine the numerators over the common denominator.

$y' = \frac{- 7 x - y \cdot 3}{3 (x + 3 y)}$

Step 5.3.3.5

Multiply $3$ by $- 1$ .

$y' = \frac{- 7 x - 3 y}{3 (x + 3 y)}$

Step 5.3.3.6

Factor $- 1$ out of $- 7 x$ .

$y' = \frac{- (7 x) - 3 y}{3 (x + 3 y)}$

Step 5.3.3.7

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

$y' = \frac{- (7 x) - (3 y)}{3 (x + 3 y)}$

Step 5.3.3.8

Factor $- 1$ out of $- (7 x) - (3 y)$ .

$y' = \frac{- (7 x + 3 y)}{3 (x + 3 y)}$

Step 5.3.3.9

Simplify the expression.

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Step 5.3.3.9.1

Rewrite $- (7 x + 3 y)$ as $- 1 (7 x + 3 y)$ .

$y' = \frac{- 1 (7 x + 3 y)}{3 (x + 3 y)}$

Step 5.3.3.9.2

Move the negative in front of the fraction.

$y' = - \frac{7 x + 3 y}{3 (x + 3 y)}$

Step 6

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

$\frac{d y}{d x} = - \frac{7 x + 3 y}{3 (x + 3 y)}$

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Set the numerator equal to zero.

$7 x + 3 y = 0$

Step 7.2

Solve the equation for $x$ .

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Step 7.2.1

Subtract $3 y$ from both sides of the equation.

$7 x = - 3 y$

Step 7.2.2

Divide each term in $7 x = - 3 y$ by $7$ and simplify.

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Step 7.2.2.1

Divide each term in $7 x = - 3 y$ by $7$ .

$\frac{7 x}{7} = \frac{- 3 y}{7}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $7$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 3 y}{7}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3 y}{7}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3 y}{7}$

Step 8

Solve for $y$ .

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Step 8.1

Simplify $7 {(- \frac{3 y}{7})}^{2} + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2}$ .

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.1.1.1.1

Apply the product rule to $- \frac{3 y}{7}$ .

$7 ({(- 1)}^{2} {(\frac{3 y}{7})}^{2}) + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.1.2

Apply the product rule to $\frac{3 y}{7}$ .

$7 ({(- 1)}^{2} \frac{{(3 y)}^{2}}{7^{2}}) + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.1.3

Apply the product rule to $3 y$ .

$7 ({(- 1)}^{2} \frac{3^{2} y^{2}}{7^{2}}) + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.2

Raise $- 1$ to the power of $2$ .

$7 (1 \frac{3^{2} y^{2}}{7^{2}}) + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.3

Multiply $\frac{3^{2} y^{2}}{7^{2}}$ by $1$ .

$7 \frac{3^{2} y^{2}}{7^{2}} + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.4

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

$7 \frac{9 y^{2}}{7^{2}} + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.5

Raise $7$ to the power of $2$ .

$7 \frac{9 y^{2}}{49} + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.6

Cancel the common factor of $7$ .

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Step 8.1.1.6.1

Factor $7$ out of $49$ .

$7 \frac{9 y^{2}}{7 (7)} + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.6.2

Cancel the common factor.

$7 \frac{9 y^{2}}{7 \cdot 7} + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.6.3

Rewrite the expression.

$\frac{9 y^{2}}{7} + 6 (- \frac{3 y}{7}) \cdot y + 9 y^{2} = 0$

Step 8.1.1.7

Multiply $6 (- \frac{3 y}{7})$ .

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Step 8.1.1.7.1

Multiply $- 1$ by $6$ .

$\frac{9 y^{2}}{7} - 6 \frac{3 y}{7} \cdot y + 9 y^{2} = 0$

Step 8.1.1.7.2

Combine $- 6$ and $\frac{3 y}{7}$ .

$\frac{9 y^{2}}{7} + \frac{- 6 (3 y)}{7} \cdot y + 9 y^{2} = 0$

Step 8.1.1.7.3

Multiply $3$ by $- 6$ .

$\frac{9 y^{2}}{7} + \frac{- 18 y}{7} \cdot y + 9 y^{2} = 0$

Step 8.1.1.8

Move the negative in front of the fraction.

$\frac{9 y^{2}}{7} - \frac{18 y}{7} \cdot y + 9 y^{2} = 0$

Step 8.1.1.9

Multiply $- \frac{18 y}{7} y$ .

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Step 8.1.1.9.1

Combine $y$ and $\frac{18 y}{7}$ .

$\frac{9 y^{2}}{7} - \frac{y (18 y)}{7} + 9 y^{2} = 0$

Step 8.1.1.9.2

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

$\frac{9 y^{2}}{7} - \frac{18 (y^{1} y)}{7} + 9 y^{2} = 0$

Step 8.1.1.9.3

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

$\frac{9 y^{2}}{7} - \frac{18 (y^{1} y^{1})}{7} + 9 y^{2} = 0$

Step 8.1.1.9.4

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

$\frac{9 y^{2}}{7} - \frac{18 y^{1 + 1}}{7} + 9 y^{2} = 0$

Step 8.1.1.9.5

Add $1$ and $1$ .

$\frac{9 y^{2}}{7} - \frac{18 y^{2}}{7} + 9 y^{2} = 0$

Step 8.1.2

Simplify terms.

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Step 8.1.2.1

Combine the numerators over the common denominator.

$9 y^{2} + \frac{9 y^{2} - 18 y^{2}}{7} = 0$

Step 8.1.2.2

Subtract $18 y^{2}$ from $9 y^{2}$ .

$9 y^{2} + \frac{- 9 y^{2}}{7} = 0$

Step 8.1.2.3

Move the negative in front of the fraction.

$9 y^{2} - \frac{9 y^{2}}{7} = 0$

Step 8.1.3

To write $9 y^{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$9 y^{2} \cdot \frac{7}{7} - \frac{9 y^{2}}{7} = 0$

Step 8.1.4

Simplify terms.

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Step 8.1.4.1

Combine $9 y^{2}$ and $\frac{7}{7}$ .

$\frac{9 y^{2} \cdot 7}{7} - \frac{9 y^{2}}{7} = 0$

Step 8.1.4.2

Combine the numerators over the common denominator.

$\frac{9 y^{2} \cdot 7 - 9 y^{2}}{7} = 0$

Step 8.1.5

Simplify the numerator.

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Step 8.1.5.1

Factor $9 y^{2}$ out of $9 y^{2} \cdot 7 - 9 y^{2}$ .

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Step 8.1.5.1.1

Factor $9 y^{2}$ out of $9 y^{2} \cdot 7$ .

$\frac{9 y^{2} (7) - 9 y^{2}}{7} = 0$

Step 8.1.5.1.2

Factor $9 y^{2}$ out of $- 9 y^{2}$ .

$\frac{9 y^{2} (7) + 9 y^{2} (- 1)}{7} = 0$

Step 8.1.5.1.3

Factor $9 y^{2}$ out of $9 y^{2} (7) + 9 y^{2} (- 1)$ .

$\frac{9 y^{2} (7 - 1)}{7} = 0$

Step 8.1.5.2

Subtract $1$ from $7$ .

$\frac{9 y^{2} \cdot 6}{7} = 0$

Step 8.1.5.3

Multiply $6$ by $9$ .

$\frac{54 y^{2}}{7} = 0$

Step 8.2

Set the numerator equal to zero.

$54 y^{2} = 0$

Step 8.3

Solve the equation for $y$ .

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Step 8.3.1

Divide each term in $54 y^{2} = 0$ by $54$ and simplify.

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Step 8.3.1.1

Divide each term in $54 y^{2} = 0$ by $54$ .

$\frac{54 y^{2}}{54} = \frac{0}{54}$

Step 8.3.1.2

Simplify the left side.

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Step 8.3.1.2.1

Cancel the common factor of $54$ .

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Step 8.3.1.2.1.1

Cancel the common factor.

$\frac{54 y^{2}}{54} = \frac{0}{54}$

Step 8.3.1.2.1.2

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

$y^{2} = \frac{0}{54}$

Step 8.3.1.3

Simplify the right side.

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Step 8.3.1.3.1

Divide $0$ by $54$ .

$y^{2} = 0$

Step 8.3.2

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

$y = \pm \sqrt{0}$

Step 8.3.3

Simplify $\pm \sqrt{0}$ .

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Step 8.3.3.1

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

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

Step 8.3.3.2

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

$y = \pm 0$

Step 8.3.3.3

Plus or minus $0$ is $0$ .

$y = 0$

Step 9

Simplify $- \frac{3 (0)}{7}$ .

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Step 9.1

Cancel the common factor of $0$ and $7$ .

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Step 9.1.1

Factor $7$ out of $3 (0)$ .

$x = - \frac{7 (3 \cdot (0))}{7}$

Step 9.1.2

Cancel the common factors.

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Step 9.1.2.1

Factor $7$ out of $7$ .

$x = - \frac{7 (3 \cdot (0))}{7 (1)}$

Step 9.1.2.2

Cancel the common factor.

$x = - \frac{7 (3 \cdot (0))}{7 \cdot 1}$

Step 9.1.2.3

Rewrite the expression.

$x = - \frac{3 \cdot (0)}{1}$

Step 9.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$x = - (3 \cdot (0))$

Step 9.2

Multiply $- (3 \cdot (0))$ .

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Step 9.2.1

Multiply $3$ by $0$ .

$x = - 0$

Step 9.2.2

Multiply $- 1$ by $0$ .

$x = 0$

Step 10

Find the points where $\frac{d y}{d x} = 0$ .

$(0, 0)$

Step 11