Calculus Examples

$10 \sin (x y) = 2 x + 3 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (10 \sin (x y)) = \frac{d}{d x} (2 x + 3 y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 \sin (x y)$ with respect to $x$ is $10 \frac{d}{d x} [\sin (x y)]$ .

$10 \frac{d}{d x} [\sin (x y)]$

Step 2.2

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

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Step 2.2.1

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

$10 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x y])$

Step 2.2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$10 (\cos (u) \frac{d}{d x} [x y])$

Step 2.2.3

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

$10 (\cos (x y) \frac{d}{d x} [x y])$

Step 2.3

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 $g (x) = y$ .

$10 \cos (x y) (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 2.4

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

$10 \cos (x y) (x y' + y \frac{d}{d x} [x])$

Step 2.5

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

$10 \cos (x y) (x y' + y \cdot 1)$

Step 2.6

Multiply $y$ by $1$ .

$10 \cos (x y) (x y' + y)$

Step 2.7

Simplify.

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Step 2.7.1

Apply the distributive property.

$10 \cos (x y) (x y') + 10 \cos (x y) y$

Step 2.7.2

Reorder terms.

$10 x \cos (x y) y' + 10 y \cos (x y)$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $2 x + 3 y$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [3 y]$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $2$ by $1$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

$2 + 3 y'$

Step 3.4

Reorder terms.

$3 y' + 2$

Step 4

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

$10 x \cos (x y) y' + 10 y \cos (x y) = 3 y' + 2$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in $10 x \cos (x y) y' + 10 y \cos (x y)$ .

$10 x y' \cos (x y) + 10 y \cos (x y) = 3 y' + 2$

Step 5.2

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

$10 x y' \cos (x y) + 10 y \cos (x y) - 3 y' = 2$

Step 5.3

Subtract $10 y \cos (x y)$ from both sides of the equation.

$10 x y' \cos (x y) - 3 y' = 2 - 10 y \cos (x y)$

Step 5.4

Factor $y'$ out of $10 x y' \cos (x y) - 3 y'$ .

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Step 5.4.1

Factor $y'$ out of $10 x y' \cos (x y)$ .

$y' (10 x \cos (x y)) - 3 y' = 2 - 10 y \cos (x y)$

Step 5.4.2

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

$y' (10 x \cos (x y)) + y' \cdot - 3 = 2 - 10 y \cos (x y)$

Step 5.4.3

Factor $y'$ out of $y' (10 x \cos (x y)) + y' \cdot - 3$ .

$y' (10 x \cos (x y) - 3) = 2 - 10 y \cos (x y)$

Step 5.5

Divide each term in $y' (10 x \cos (x y) - 3) = 2 - 10 y \cos (x y)$ by $10 x \cos (x y) - 3$ and simplify.

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Step 5.5.1

Divide each term in $y' (10 x \cos (x y) - 3) = 2 - 10 y \cos (x y)$ by $10 x \cos (x y) - 3$ .

$\frac{y' (10 x \cos (x y) - 3)}{10 x \cos (x y) - 3} = \frac{2}{10 x \cos (x y) - 3} + \frac{- 10 y \cos (x y)}{10 x \cos (x y) - 3}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $10 x \cos (x y) - 3$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{y' (10 x \cos (x y) - 3)}{10 x \cos (x y) - 3} = \frac{2}{10 x \cos (x y) - 3} + \frac{- 10 y \cos (x y)}{10 x \cos (x y) - 3}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{2}{10 x \cos (x y) - 3} + \frac{- 10 y \cos (x y)}{10 x \cos (x y) - 3}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

$y' = \frac{2}{10 x \cos (x y) - 3} - \frac{10 y \cos (x y)}{10 x \cos (x y) - 3}$

Step 5.5.3.2

Combine the numerators over the common denominator.

$y' = \frac{2 - 10 y \cos (x y)}{10 x \cos (x y) - 3}$

Step 5.5.3.3

Factor $2$ out of $2 - 10 y \cos (x y)$ .

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Step 5.5.3.3.1

Factor $2$ out of $2$ .

$y' = \frac{2 (1) - 10 y \cos (x y)}{10 x \cos (x y) - 3}$

Step 5.5.3.3.2

Factor $2$ out of $- 10 y \cos (x y)$ .

$y' = \frac{2 (1) + 2 (- 5 y \cos (x y))}{10 x \cos (x y) - 3}$

Step 5.5.3.3.3

Factor $2$ out of $2 (1) + 2 (- 5 y \cos (x y))$ .

$y' = \frac{2 (1 - 5 y \cos (x y))}{10 x \cos (x y) - 3}$

Step 6

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

$\frac{d y}{d x} = \frac{2 (1 - 5 y \cos (x y))}{10 x \cos (x y) - 3}$

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.

$2 (1 - 5 y \cos (x y)) = 0$

Step 7.2

Solve the equation for $x$ .

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Step 7.2.1

Divide each term in $2 (1 - 5 y \cos (x y)) = 0$ by $2$ and simplify.

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Step 7.2.1.1

Divide each term in $2 (1 - 5 y \cos (x y)) = 0$ by $2$ .

$\frac{2 (1 - 5 y \cos (x y))}{2} = \frac{0}{2}$

Step 7.2.1.2

Simplify the left side.

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Step 7.2.1.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.2.1.1

Cancel the common factor.

$\frac{2 (1 - 5 y \cos (x y))}{2} = \frac{0}{2}$

Step 7.2.1.2.1.2

Divide $1 - 5 y \cos (x y)$ by $1$ .

$1 - 5 y \cos (x y) = \frac{0}{2}$

Step 7.2.1.3

Simplify the right side.

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Step 7.2.1.3.1

Divide $0$ by $2$ .

$1 - 5 y \cos (x y) = 0$

Step 7.2.2

Subtract $1$ from both sides of the equation.

$- 5 y \cos (x y) = - 1$

Step 7.2.3

Divide each term in $- 5 y \cos (x y) = - 1$ by $- 5 y$ and simplify.

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Step 7.2.3.1

Divide each term in $- 5 y \cos (x y) = - 1$ by $- 5 y$ .

$\frac{- 5 y \cos (x y)}{- 5 y} = \frac{- 1}{- 5 y}$

Step 7.2.3.2

Simplify the left side.

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Step 7.2.3.2.1

Cancel the common factor of $- 5$ .

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Step 7.2.3.2.1.1

Cancel the common factor.

$\frac{- 5 y \cos (x y)}{- 5 y} = \frac{- 1}{- 5 y}$

Step 7.2.3.2.1.2

Rewrite the expression.

$\frac{y \cos (x y)}{y} = \frac{- 1}{- 5 y}$

Step 7.2.3.2.2

Cancel the common factor of $y$ .

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Step 7.2.3.2.2.1

Cancel the common factor.

$\frac{y \cos (x y)}{y} = \frac{- 1}{- 5 y}$

Step 7.2.3.2.2.2

Divide $\cos (x y)$ by $1$ .

$\cos (x y) = \frac{- 1}{- 5 y}$

Step 7.2.3.3

Simplify the right side.

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Step 7.2.3.3.1

Dividing two negative values results in a positive value.

$\cos (x y) = \frac{1}{5 y}$

Step 7.2.4

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x y = \arccos (\frac{1}{5 y})$

Step 7.2.5

Divide each term in $x y = \arccos (\frac{1}{5 y})$ by $y$ and simplify.

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Step 7.2.5.1

Divide each term in $x y = \arccos (\frac{1}{5 y})$ by $y$ .

$\frac{x y}{y} = \frac{\arccos (\frac{1}{5 y})}{y}$

Step 7.2.5.2

Simplify the left side.

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Step 7.2.5.2.1

Cancel the common factor of $y$ .

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Step 7.2.5.2.1.1

Cancel the common factor.

$\frac{x y}{y} = \frac{\arccos (\frac{1}{5 y})}{y}$

Step 7.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arccos (\frac{1}{5 y})}{y}$

Step 8

Solve for $y$ .

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Step 8.1

Simplify $10 \sin ((\frac{\arccos (\frac{1}{5 y})}{y}) y)$ .

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Step 8.1.1

Reduce the expression by cancelling the common factors.

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Step 8.1.1.1

Cancel the common factor of $y$ .

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Step 8.1.1.1.1

Cancel the common factor.

$10 \sin (\frac{\arccos (\frac{1}{5 y})}{y} y) = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.1.1.2

Rewrite the expression.

$10 \sin (\arccos (\frac{1}{5 y})) = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.1.2

Write the expression using exponents.

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Step 8.1.1.2.1

Draw a triangle in the plane with vertices $(\frac{1}{5 y}, \sqrt{1^{2} - {(\frac{1}{5 y})}^{2}})$ , $(\frac{1}{5 y}, 0)$ , and the origin. Then $\arccos (\frac{1}{5 y})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\frac{1}{5 y}, \sqrt{1^{2} - {(\frac{1}{5 y})}^{2}})$ . Therefore, $\sin (\arccos (\frac{1}{5 y}))$ is $\sqrt{1 - {(\frac{1}{5 y})}^{2}}$ .

$10 \sqrt{1 - {(\frac{1}{5 y})}^{2}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.1.2.2

Rewrite $1$ as $1^{2}$ .

$10 \sqrt{1^{2} - {(\frac{1}{5 y})}^{2}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{1}{5 y}$ .

$10 \sqrt{(1 + \frac{1}{5 y}) (1 - \frac{1}{5 y})} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.3

Simplify terms.

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Step 8.1.3.1

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

$10 \sqrt{(\frac{5 y}{5 y} + \frac{1}{5 y}) (1 - \frac{1}{5 y})} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.3.2

Combine the numerators over the common denominator.

$10 \sqrt{\frac{5 y + 1}{5 y} (1 - \frac{1}{5 y})} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.3.3

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

$10 \sqrt{\frac{5 y + 1}{5 y} (\frac{5 y}{5 y} - \frac{1}{5 y})} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.3.4

Combine the numerators over the common denominator.

$10 \sqrt{\frac{5 y + 1}{5 y} \cdot \frac{5 y - 1}{5 y}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.3.5

Multiply $\frac{5 y + 1}{5 y}$ by $\frac{5 y - 1}{5 y}$ .

$10 \sqrt{\frac{(5 y + 1) (5 y - 1)}{5 y (5 y)}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.4

Combine exponents.

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Step 8.1.4.1

Multiply $5$ by $5$ .

$10 \sqrt{\frac{(5 y + 1) (5 y - 1)}{25 y \cdot y}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.4.2

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

$10 \sqrt{\frac{(5 y + 1) (5 y - 1)}{25 (y^{1} y)}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.4.3

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

$10 \sqrt{\frac{(5 y + 1) (5 y - 1)}{25 (y^{1} y^{1})}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.4.4

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

$10 \sqrt{\frac{(5 y + 1) (5 y - 1)}{25 y^{1 + 1}}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.4.5

Add $1$ and $1$ .

$10 \sqrt{\frac{(5 y + 1) (5 y - 1)}{25 y^{2}}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.5

Rewrite $\frac{(5 y + 1) (5 y - 1)}{25 y^{2}}$ as ${(\frac{1}{5 y})}^{2} ((5 y + 1) (5 y - 1))$ .

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Step 8.1.5.1

Factor the perfect power $1^{2}$ out of $(5 y + 1) (5 y - 1)$ .

$10 \sqrt{\frac{1^{2} ((5 y + 1) (5 y - 1))}{25 y^{2}}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.5.2

Factor the perfect power ${(5 y)}^{2}$ out of $25 y^{2}$ .

$10 \sqrt{\frac{1^{2} ((5 y + 1) (5 y - 1))}{{(5 y)}^{2} \cdot 1}} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.5.3

Rearrange the fraction $\frac{1^{2} ((5 y + 1) (5 y - 1))}{{(5 y)}^{2} \cdot 1}$ .

$10 \sqrt{{(\frac{1}{5 y})}^{2} ((5 y + 1) (5 y - 1))} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.6

Pull terms out from under the radical.

$10 (\frac{1}{5 y} \sqrt{(5 y + 1) (5 y - 1)}) = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.7

Combine $\frac{1}{5 y}$ and $\sqrt{(5 y + 1) (5 y - 1)}$ .

$10 \frac{\sqrt{(5 y + 1) (5 y - 1)}}{5 y} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.8

Cancel the common factor of $5$ .

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Step 8.1.8.1

Factor $5$ out of $10$ .

$5 (2) \frac{\sqrt{(5 y + 1) (5 y - 1)}}{5 y} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.8.2

Factor $5$ out of $5 y$ .

$5 (2) \frac{\sqrt{(5 y + 1) (5 y - 1)}}{5 (y)} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.8.3

Cancel the common factor.

$5 \cdot 2 \frac{\sqrt{(5 y + 1) (5 y - 1)}}{5 y} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.8.4

Rewrite the expression.

$2 \frac{\sqrt{(5 y + 1) (5 y - 1)}}{y} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.1.9

Combine $2$ and $\frac{\sqrt{(5 y + 1) (5 y - 1)}}{y}$ .

$\frac{2 \sqrt{(5 y + 1) (5 y - 1)}}{y} = 2 (\frac{\arccos (\frac{1}{5 y})}{y}) + 3 y$

Step 8.2

Combine $2$ and $\frac{\arccos (\frac{1}{5 y})}{y}$ .

$\frac{2 \sqrt{(5 y + 1) (5 y - 1)}}{y} = \frac{2 \arccos (\frac{1}{5 y})}{y} + 3 y$

Step 8.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$y \approx 0.24419009, 2.99063055$

Step 9

Solve for $x$ when $y$ is $0.24419009$ .

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Step 9.1

Remove parentheses.

$x = \frac{\arccos (\frac{1}{5 (0.24419009)})}{0.24419009}$

Step 9.2

Simplify $\frac{\arccos (\frac{1}{5 (0.24419009)})}{0.24419009}$ .

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Step 9.2.1

Multiply $5$ by $0.24419009$ .

$x = \frac{\arccos (\frac{1}{1.22095048})}{0.24419009}$

Step 9.2.2

Simplify the numerator.

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Step 9.2.2.1

Divide $1$ by $1.22095048$ .

$x = \frac{\arccos (0.81903403)}{0.24419009}$

Step 9.2.2.2

Evaluate $\arccos (0.81903403)$ .

$x = \frac{0.61107095}{0.24419009}$

Step 9.2.3

Divide $0.61107095$ by $0.24419009$ .

$x = 2.50243954$

Step 10

Solve for $x$ when $y$ is $2.99063055$ .

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Step 10.1

Remove parentheses.

$x = \frac{\arccos (\frac{1}{5 (2.99063055)})}{2.99063055}$

Step 10.2

Simplify $\frac{\arccos (\frac{1}{5 (2.99063055)})}{2.99063055}$ .

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Step 10.2.1

Multiply $5$ by $2.99063055$ .

$x = \frac{\arccos (\frac{1}{14.95315279})}{2.99063055}$

Step 10.2.2

Simplify the numerator.

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Step 10.2.2.1

Divide $1$ by $14.95315279$ .

$x = \frac{\arccos (0.06687552)}{2.99063055}$

Step 10.2.2.2

Evaluate $\arccos (0.06687552)$ .

$x = \frac{1.50387084}{2.99063055}$

Step 10.2.3

Divide $1.50387084$ by $2.99063055$ .

$x = 0.50286079$

Step 11

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

$(2.50243954, 0.24419009)$

$(0.50286079, 2.99063055)$

Step 12