Calculus Examples

$y = 6 x - 4 \cos (3 x)$

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (6 x - 4 \cos (3 x))$

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Differentiate the right side of the equation.

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By the Sum Rule, the derivative of $6 x - 4 \cos (3 x)$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 4 \cos (3 x)]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [- 4 \cos (3 x)]$

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

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

$6 \frac{d}{d x} [x] + \frac{d}{d x} [- 4 \cos (3 x)]$

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

$6 \cdot 1 + \frac{d}{d x} [- 4 \cos (3 x)]$

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [- 4 \cos (3 x)]$

Evaluate $\frac{d}{d x} [- 4 \cos (3 x)]$ .

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Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \cos (3 x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\cos (3 x)]$ .

$6 - 4 \frac{d}{d x} [\cos (3 x)]$

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

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To apply the Chain Rule, set $u$ as $3 x$ .

$6 - 4 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [3 x])$

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

$6 - 4 (- \sin (u) \frac{d}{d x} [3 x])$

Replace all occurrences of $u$ with $3 x$ .

$6 - 4 (- \sin (3 x) \frac{d}{d x} [3 x])$

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

$6 - 4 (- \sin (3 x) (3 \frac{d}{d x} [x]))$

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

$6 - 4 (- \sin (3 x) (3 \cdot 1))$

Multiply $3$ by $1$ .

$6 - 4 (- \sin (3 x) \cdot 3)$

Multiply $3$ by $- 1$ .

$6 - 4 (- 3 \sin (3 x))$

Multiply $- 3$ by $- 4$ .

$6 + 12 \sin (3 x)$

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

$y' = 6 + 12 \sin (3 x)$

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

$\frac{d y}{d x} = 6 + 12 \sin (3 x)$

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

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Rewrite the equation as $6 + 12 \sin (3 x) = 0$ .

$6 + 12 \sin (3 x) = 0$

Subtract $6$ from both sides of the equation.

$12 \sin (3 x) = - 6$

Divide each term by $12$ and simplify.

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Divide each term in $12 \sin (3 x) = - 6$ by $12$ .

$\frac{12 \sin (3 x)}{12} = \frac{- 6}{12}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{12 \sin (3 x)}{12} = \frac{- 6}{12}$

Divide $\sin (3 x)$ by $1$ .

$\sin (3 x) = \frac{- 6}{12}$

Simplify $\frac{- 6}{12}$ .

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Reduce the expression by cancelling the common factors.

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Factor $6$ out of $- 6$ .

$\sin (3 x) = \frac{6 (- 1)}{12}$

Cancel the common factors.

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Factor $6$ out of $12$ .

$\sin (3 x) = \frac{6 \cdot - 1}{6 \cdot 2}$

Cancel the common factor.

$\sin (3 x) = \frac{6 \cdot - 1}{6 \cdot 2}$

Rewrite the expression.

$\sin (3 x) = \frac{- 1}{2}$

Move the negative in front of the fraction.

$\sin (3 x) = - \frac{1}{2}$

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

$3 x = \arcsin (- \frac{1}{2})$

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$3 x = - \frac{π}{6}$

Divide each term by $3$ and simplify.

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Divide each term in $3 x = - \frac{π}{6}$ by $3$ .

$\frac{3 x}{3} = - \frac{π}{6} \cdot \frac{1}{3}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{3 x}{3} = - \frac{π}{6} \cdot \frac{1}{3}$

Divide $x$ by $1$ .

$x = - \frac{π}{6} \cdot \frac{1}{3}$

Multiply $- \frac{π}{6} \cdot \frac{1}{3}$ .

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Multiply $\frac{1}{3}$ and $\frac{π}{6}$ .

$x = - \frac{π}{3 \cdot 6}$

Multiply $3$ by $6$ .

$x = - \frac{π}{18}$

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$3 x = 2 π + \frac{π}{6} + π$

Simplify the expression to find the second solution.

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$3 x = \frac{2 π}{1} \cdot \frac{6}{6} + \frac{π}{6} + π$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 π \cdot 6}{1 \cdot 6} + \frac{π}{6} + π$

Multiply $6$ by $1$ .

$\frac{2 π \cdot 6}{6} + \frac{π}{6} + π$

$3 x = \frac{2 π \cdot 6}{6} + \frac{π}{6} + π$

Combine the numerators over the common denominator.

$3 x = \frac{2 π \cdot 6 + π}{6} + π$

To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$3 x = \frac{2 π \cdot 6 + π}{6} + \frac{π}{1} \cdot \frac{6}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 π \cdot 6 + π}{6} + \frac{π \cdot 6}{1 \cdot 6}$

Multiply $6$ by $1$ .

$\frac{2 π \cdot 6 + π}{6} + \frac{π \cdot 6}{6}$

$3 x = \frac{2 π \cdot 6 + π}{6} + \frac{π \cdot 6}{6}$

Combine the numerators over the common denominator.

$3 x = \frac{2 π \cdot 6 + π + π \cdot 6}{6}$

Add $2 π \cdot 6$ and $π \cdot 6$ .

$3 x = \frac{3 π \cdot 6 + π}{6}$

Multiply $6$ by $3$ .

$3 x = \frac{18 π + π}{6}$

Add $18 π$ and $π$ .

$3 x = \frac{19 π}{6}$

Divide each term by $3$ and simplify.

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Divide each term in $3 x = \frac{19 π}{6}$ by $3$ .

$\frac{3 x}{3} = \frac{19 π}{6} \cdot \frac{1}{3}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{3 x}{3} = \frac{19 π}{6} \cdot \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{19 π}{6} \cdot \frac{1}{3}$

Multiply $\frac{19 π}{6} \cdot \frac{1}{3}$ .

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Multiply $\frac{19 π}{6}$ and $\frac{1}{3}$ .

$x = \frac{19 π}{6 \cdot 3}$

Multiply $6$ by $3$ .

$x = \frac{19 π}{18}$

Find the period.

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The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Add $\frac{2 π}{3}$ to every negative angle to get positive angles.

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Add $\frac{2 π}{3}$ to $- \frac{π}{18}$ to find the positive angle.

$- \frac{π}{18} + \frac{2 π}{3}$

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{2 π}{3} \cdot \frac{6}{6} - \frac{π}{18}$

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 π \cdot 6}{3 \cdot 6} - \frac{π}{18}$

Multiply $3$ by $6$ .

$\frac{2 π \cdot 6}{18} - \frac{π}{18}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{18}$

Simplify the numerator.

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Multiply $6$ by $2$ .

$\frac{12 π - π}{18}$

Subtract $π$ from $12 π$ .

$\frac{11 π}{18}$

List the new angles.

$x = \frac{11 π}{18}$

The period of the $\sin (3 x)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = \frac{11 π}{18} + \frac{2 π n}{3}, \frac{19 π}{18} + \frac{2 π n}{3}$ , for any integer $n$

Simplify $6 (\frac{11 π}{18} + \frac{2 π n}{3}) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$ .

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Simplify each term.

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Apply the distributive property.

$y = 6 (\frac{11 π}{18}) + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Cancel the common factor of $6$ .

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Write $6$ as a fraction with denominator $1$ .

$y = \frac{6}{1} \cdot \frac{11 π}{18} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Factor out the greatest common factor $6$ .

$y = \frac{6 \cdot 1}{1} \cdot \frac{11 π}{6 \cdot 3} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Cancel the common factor.

$y = \frac{6 \cdot 1}{1} \cdot \frac{11 π}{6 \cdot 3} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Rewrite the expression.

$y = \frac{1}{1} \cdot \frac{11 π}{3} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Multiply $\frac{1}{1}$ and $\frac{11 π}{3}$ .

$y = \frac{11 π}{3} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Cancel the common factor of $3$ .

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Write $6$ as a fraction with denominator $1$ .

$y = \frac{11 π}{3} + \frac{6}{1} \cdot \frac{2 π n}{3} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Factor out the greatest common factor $3$ .

$y = \frac{11 π}{3} + \frac{3 \cdot 2}{1} \cdot \frac{2 π n}{3 \cdot 1} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Cancel the common factor.

$y = \frac{11 π}{3} + \frac{3 \cdot 2}{1} \cdot \frac{2 π n}{3 \cdot 1} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Rewrite the expression.

$y = \frac{11 π}{3} + \frac{2}{1} \cdot \frac{2 π n}{1} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Simplify.

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Multiply $\frac{2}{1}$ and $\frac{2 π n}{1}$ .

$y = \frac{11 π}{3} + \frac{2 (2 π n)}{1} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Multiply $2$ by $2$ .

$y = \frac{11 π}{3} + \frac{4 (π n)}{1} - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Divide $4 (π n)$ by $1$ .

$y = \frac{11 π}{3} + 4 (π n) - 4 \cos (3 (\frac{11 π}{18} + \frac{2 π n}{3}))$

Apply the distributive property.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (3 (\frac{11 π}{18}) + 3 (\frac{2 π n}{3}))$

Cancel the common factor of $3$ .

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Write $3$ as a fraction with denominator $1$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{3}{1} \cdot \frac{11 π}{18} + 3 (\frac{2 π n}{3}))$

Factor out the greatest common factor $3$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{3 \cdot 1}{1} \cdot \frac{11 π}{3 \cdot 6} + 3 (\frac{2 π n}{3}))$

Cancel the common factor.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{3 \cdot 1}{1} \cdot \frac{11 π}{3 \cdot 6} + 3 (\frac{2 π n}{3}))$

Rewrite the expression.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{1}{1} \cdot \frac{11 π}{6} + 3 (\frac{2 π n}{3}))$

Multiply $\frac{1}{1}$ and $\frac{11 π}{6}$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + 3 (\frac{2 π n}{3}))$

Cancel the common factor of $3$ .

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Write $3$ as a fraction with denominator $1$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + \frac{3}{1} \cdot \frac{2 π n}{3})$

Factor out the greatest common factor $3$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + \frac{3 \cdot 1}{1} \cdot \frac{2 π n}{3 \cdot 1})$

Cancel the common factor.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + \frac{3 \cdot 1}{1} \cdot \frac{2 π n}{3 \cdot 1})$

Rewrite the expression.

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + \frac{1}{1} \cdot \frac{2 π n}{1})$

Simplify.

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Multiply $\frac{1}{1}$ and $\frac{2 π n}{1}$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + \frac{2 π n}{1})$

Divide $2 π n$ by $1$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (\frac{11 π}{6} + 2 π n)$

Simplify with commuting.

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Reorder $\frac{11 π}{6}$ and $2 π n$ .

$y = \frac{11 π}{3} + 4 π n - 4 \cos (2 π n + \frac{11 π}{6})$

Reorder $\frac{11 π}{3}$ and $4 π n$ .

$y = 4 π n + \frac{11 π}{3} - 4 \cos (2 π n + \frac{11 π}{6})$

Simplify $6 (\frac{19 π}{18} + \frac{2 π n}{3}) - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$ .

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Simplify each term.

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Apply the distributive property.

$y = 6 (\frac{19 π}{18}) + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Cancel the common factor of $6$ .

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Write $6$ as a fraction with denominator $1$ .

$y = \frac{6}{1} \cdot \frac{19 π}{18} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Factor out the greatest common factor $6$ .

$y = \frac{6 \cdot 1}{1} \cdot \frac{19 π}{6 \cdot 3} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Cancel the common factor.

$y = \frac{6 \cdot 1}{1} \cdot \frac{19 π}{6 \cdot 3} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Rewrite the expression.

$y = \frac{1}{1} \cdot \frac{19 π}{3} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Multiply $\frac{1}{1}$ and $\frac{19 π}{3}$ .

$y = \frac{19 π}{3} + 6 (\frac{2 π n}{3}) - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Cancel the common factor of $3$ .

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Write $6$ as a fraction with denominator $1$ .

$y = \frac{19 π}{3} + \frac{6}{1} \cdot \frac{2 π n}{3} - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Factor out the greatest common factor $3$ .

$y = \frac{19 π}{3} + \frac{3 \cdot 2}{1} \cdot \frac{2 π n}{3 \cdot 1} - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Cancel the common factor.

$y = \frac{19 π}{3} + \frac{3 \cdot 2}{1} \cdot \frac{2 π n}{3 \cdot 1} - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Rewrite the expression.

$y = \frac{19 π}{3} + \frac{2}{1} \cdot \frac{2 π n}{1} - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Simplify.

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Multiply $\frac{2}{1}$ and $\frac{2 π n}{1}$ .

$y = \frac{19 π}{3} + \frac{2 (2 π n)}{1} - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Multiply $2$ by $2$ .

$y = \frac{19 π}{3} + \frac{4 (π n)}{1} - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Divide $4 (π n)$ by $1$ .

$y = \frac{19 π}{3} + 4 (π n) - 4 \cos (3 (\frac{19 π}{18} + \frac{2 π n}{3}))$

Apply the distributive property.

$y = \frac{19 π}{3} + 4 π n - 4 \cos (3 (\frac{19 π}{18}) + 3 (\frac{2 π n}{3}))$

Cancel the common factor of $3$ .

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Write $3$ as a fraction with denominator $1$ .

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{3}{1} \cdot \frac{19 π}{18} + 3 (\frac{2 π n}{3}))$

Factor out the greatest common factor $3$ .

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{3 \cdot 1}{1} \cdot \frac{19 π}{3 \cdot 6} + 3 (\frac{2 π n}{3}))$

Cancel the common factor.

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{3 \cdot 1}{1} \cdot \frac{19 π}{3 \cdot 6} + 3 (\frac{2 π n}{3}))$

Rewrite the expression.

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{1}{1} \cdot \frac{19 π}{6} + 3 (\frac{2 π n}{3}))$

Multiply $\frac{1}{1}$ and $\frac{19 π}{6}$ .

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{19 π}{6} + 3 (\frac{2 π n}{3}))$

Cancel the common factor of $3$ .

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Write $3$ as a fraction with denominator $1$ .

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{19 π}{6} + \frac{3}{1} \cdot \frac{2 π n}{3})$

Factor out the greatest common factor $3$ .

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{19 π}{6} + \frac{3 \cdot 1}{1} \cdot \frac{2 π n}{3 \cdot 1})$

Cancel the common factor.

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{19 π}{6} + \frac{3 \cdot 1}{1} \cdot \frac{2 π n}{3 \cdot 1})$

Rewrite the expression.

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{19 π}{6} + \frac{1}{1} \cdot \frac{2 π n}{1})$

Simplify.

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Multiply $\frac{1}{1}$ and $\frac{2 π n}{1}$ .

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{19 π}{6} + \frac{2 π n}{1})$

Divide $2 π n$ by $1$ .

$y = \frac{19 π}{3} + 4 π n - 4 \cos (\frac{19 π}{6} + 2 π n)$

Simplify with commuting.

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Reorder $\frac{19 π}{6}$ and $2 π n$ .

$y = \frac{19 π}{3} + 4 π n - 4 \cos (2 π n + \frac{19 π}{6})$

Reorder $\frac{19 π}{3}$ and $4 π n$ .

$y = 4 π n + \frac{19 π}{3} - 4 \cos (2 π n + \frac{19 π}{6})$

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

$(\frac{11 π}{18} + \frac{2 π n}{3}, 4 π n + \frac{11 π}{3} - 4 \cos (2 π n + \frac{11 π}{6}))$ , for any integer $n$

$(\frac{19 π}{18} + \frac{2 π n}{3}, 4 π n + \frac{19 π}{3} - 4 \cos (2 π n + \frac{19 π}{6}))$ , for any integer $n$

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