Calculus Examples

$\tan (4 x + y) = 4 x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\tan (4 x + y)) = \frac{d}{d x} (4 x)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 4 x + y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $4 x + y$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d x} [4 x + y]$

Step 2.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\sec^{2} (u) \frac{d}{d x} [4 x + y]$

Step 2.1.3

Replace all occurrences of $u$ with $4 x + y$ .

$\sec^{2} (4 x + y) \frac{d}{d x} [4 x + y]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $4 x + y$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [y]$ .

$\sec^{2} (4 x + y) (\frac{d}{d x} [4 x] + \frac{d}{d x} [y])$

Step 2.2.2

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

$\sec^{2} (4 x + y) (4 \frac{d}{d x} [x] + \frac{d}{d x} [y])$

Step 2.2.3

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

$\sec^{2} (4 x + y) (4 \cdot 1 + \frac{d}{d x} [y])$

Step 2.2.4

Multiply $4$ by $1$ .

$\sec^{2} (4 x + y) (4 + \frac{d}{d x} [y])$

Step 2.3

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

$\sec^{2} (4 x + y) (4 + y')$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$4 \frac{d}{d x} [x]$

Step 3.2

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

$4 \cdot 1$

Step 3.3

Multiply $4$ by $1$ .

$4$

Step 4

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

$\sec^{2} (4 x + y) (4 + y') = 4$

Step 5

Solve for $y'$ .

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Step 5.1

Divide each term in $\sec^{2} (4 x + y) (4 + y') = 4$ by $\sec^{2} (4 x + y)$ and simplify.

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Step 5.1.1

Divide each term in $\sec^{2} (4 x + y) (4 + y') = 4$ by $\sec^{2} (4 x + y)$ .

$\frac{\sec^{2} (4 x + y) (4 + y')}{\sec^{2} (4 x + y)} = \frac{4}{\sec^{2} (4 x + y)}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $\sec^{2} (4 x + y)$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{\sec^{2} (4 x + y) (4 + y')}{\sec^{2} (4 x + y)} = \frac{4}{\sec^{2} (4 x + y)}$

Step 5.1.2.1.2

Divide $4 + y'$ by $1$ .

$4 + y' = \frac{4}{\sec^{2} (4 x + y)}$

Step 5.2

Subtract $4$ from both sides of the equation.

$y' = \frac{4}{\sec^{2} (4 x + y)} - 4$

Step 6

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

$\frac{d y}{d x} = \frac{4}{\sec^{2} (4 x + y)} - 4$

Step 7

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

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Step 7.1

Add $4$ to both sides of the equation.

$\frac{4}{\sec^{2} (4 x + y)} = 4$

Step 7.2

Find the LCD of the terms in the equation.

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Step 7.2.1

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

$\sec^{2} (4 x + y), 1$

Step 7.2.2

The LCM of one and any expression is the expression.

$\sec^{2} (4 x + y)$

Step 7.3

Multiply each term in $\frac{4}{\sec^{2} (4 x + y)} = 4$ by $\sec^{2} (4 x + y)$ to eliminate the fractions.

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Step 7.3.1

Multiply each term in $\frac{4}{\sec^{2} (4 x + y)} = 4$ by $\sec^{2} (4 x + y)$ .

$\frac{4}{\sec^{2} (4 x + y)} \sec^{2} (4 x + y) = 4 \sec^{2} (4 x + y)$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $\sec^{2} (4 x + y)$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{4}{\sec^{2} (4 x + y)} \sec^{2} (4 x + y) = 4 \sec^{2} (4 x + y)$

Step 7.3.2.1.2

Rewrite the expression.

$4 = 4 \sec^{2} (4 x + y)$

Step 7.4

Solve the equation.

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Step 7.4.1

Rewrite the equation as $4 \sec^{2} (4 x + y) = 4$ .

$4 \sec^{2} (4 x + y) = 4$

Step 7.4.2

Divide each term in $4 \sec^{2} (4 x + y) = 4$ by $4$ and simplify.

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Step 7.4.2.1

Divide each term in $4 \sec^{2} (4 x + y) = 4$ by $4$ .

$\frac{4 \sec^{2} (4 x + y)}{4} = \frac{4}{4}$

Step 7.4.2.2

Simplify the left side.

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Step 7.4.2.2.1

Cancel the common factor of $4$ .

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Step 7.4.2.2.1.1

Cancel the common factor.

$\frac{4 \sec^{2} (4 x + y)}{4} = \frac{4}{4}$

Step 7.4.2.2.1.2

Divide $\sec^{2} (4 x + y)$ by $1$ .

$\sec^{2} (4 x + y) = \frac{4}{4}$

Step 7.4.2.3

Simplify the right side.

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Step 7.4.2.3.1

Divide $4$ by $4$ .

$\sec^{2} (4 x + y) = 1$

Step 7.4.3

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

$\sec (4 x + y) = \pm \sqrt{1}$

Step 7.4.4

Any root of $1$ is $1$ .

$\sec (4 x + y) = \pm 1$

Step 7.4.5

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

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Step 7.4.5.1

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

$\sec (4 x + y) = 1$

Step 7.4.5.2

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

$\sec (4 x + y) = - 1$

Step 7.4.5.3

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

$\sec (4 x + y) = 1, - 1$

Step 7.5

Set up each of the solutions to solve for $x$ .

$\sec (4 x + y) = 1$

$\sec (4 x + y) = - 1$

Step 7.6

Solve for $x$ in $\sec (4 x + y) = 1$ .

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Step 7.6.1

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

$4 x + y = arcsec (1)$

Step 7.6.2

Simplify the right side.

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Step 7.6.2.1

The exact value of $arcsec (1)$ is $0$ .

$4 x + y = 0$

Step 7.6.3

Subtract $y$ from both sides of the equation.

$4 x = - y$

Step 7.6.4

Divide each term in $4 x = - y$ by $4$ and simplify.

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Step 7.6.4.1

Divide each term in $4 x = - y$ by $4$ .

$\frac{4 x}{4} = \frac{- y}{4}$

Step 7.6.4.2

Simplify the left side.

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Step 7.6.4.2.1

Cancel the common factor of $4$ .

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Step 7.6.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- y}{4}$

Step 7.6.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y}{4}$

Step 7.6.4.3

Simplify the right side.

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Step 7.6.4.3.1

Move the negative in front of the fraction.

$x = - \frac{y}{4}$

Step 7.6.5

The secant function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$4 x + y = 2 π - 0$

Step 7.6.6

Solve for $x$ .

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Step 7.6.6.1

Subtract $0$ from $2 π$ .

$4 x + y = 2 π$

Step 7.6.6.2

Subtract $y$ from both sides of the equation.

$4 x = 2 π - y$

Step 7.6.6.3

Divide each term in $4 x = 2 π - y$ by $4$ and simplify.

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Step 7.6.6.3.1

Divide each term in $4 x = 2 π - y$ by $4$ .

$\frac{4 x}{4} = \frac{2 π}{4} + \frac{- y}{4}$

Step 7.6.6.3.2

Simplify the left side.

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Step 7.6.6.3.2.1

Cancel the common factor of $4$ .

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Step 7.6.6.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{2 π}{4} + \frac{- y}{4}$

Step 7.6.6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{4} + \frac{- y}{4}$

Step 7.6.6.3.3

Simplify the right side.

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Step 7.6.6.3.3.1

Simplify each term.

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Step 7.6.6.3.3.1.1

Cancel the common factor of $2$ and $4$ .

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Step 7.6.6.3.3.1.1.1

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{4} + \frac{- y}{4}$

Step 7.6.6.3.3.1.1.2

Cancel the common factors.

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Step 7.6.6.3.3.1.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 π}{2 \cdot 2} + \frac{- y}{4}$

Step 7.6.6.3.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 2} + \frac{- y}{4}$

Step 7.6.6.3.3.1.1.2.3

Rewrite the expression.

$x = \frac{π}{2} + \frac{- y}{4}$

Step 7.6.6.3.3.1.2

Move the negative in front of the fraction.

$x = \frac{π}{2} - \frac{y}{4}$

Step 7.7

Solve for $x$ in $\sec (4 x + y) = - 1$ .

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Step 7.7.1

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

$4 x + y = arcsec (- 1)$

Step 7.7.2

Simplify the right side.

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Step 7.7.2.1

The exact value of $arcsec (- 1)$ is $π$ .

$4 x + y = π$

Step 7.7.3

Subtract $y$ from both sides of the equation.

$4 x = π - y$

Step 7.7.4

Divide each term in $4 x = π - y$ by $4$ and simplify.

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Step 7.7.4.1

Divide each term in $4 x = π - y$ by $4$ .

$\frac{4 x}{4} = \frac{π}{4} + \frac{- y}{4}$

Step 7.7.4.2

Simplify the left side.

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Step 7.7.4.2.1

Cancel the common factor of $4$ .

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Step 7.7.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{π}{4} + \frac{- y}{4}$

Step 7.7.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{4} + \frac{- y}{4}$

Step 7.7.4.3

Simplify the right side.

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Step 7.7.4.3.1

Move the negative in front of the fraction.

$x = \frac{π}{4} - \frac{y}{4}$

Step 7.7.5

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$4 x + y = 2 π - π$

Step 7.7.6

Solve for $x$ .

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Step 7.7.6.1

Subtract $π$ from $2 π$ .

$4 x + y = π$

Step 7.7.6.2

Subtract $y$ from both sides of the equation.

$4 x = π - y$

Step 7.7.6.3

Divide each term in $4 x = π - y$ by $4$ and simplify.

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Step 7.7.6.3.1

Divide each term in $4 x = π - y$ by $4$ .

$\frac{4 x}{4} = \frac{π}{4} + \frac{- y}{4}$

Step 7.7.6.3.2

Simplify the left side.

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Step 7.7.6.3.2.1

Cancel the common factor of $4$ .

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Step 7.7.6.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{π}{4} + \frac{- y}{4}$

Step 7.7.6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{4} + \frac{- y}{4}$

Step 7.7.6.3.3

Simplify the right side.

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Step 7.7.6.3.3.1

Move the negative in front of the fraction.

$x = \frac{π}{4} - \frac{y}{4}$

Step 7.8

List all of the solutions.

$x = \frac{π}{2} - \frac{y}{4}$

$x = \frac{π}{4} - \frac{y}{4}$

$x = \frac{π}{2} - \frac{y}{4}$

$x = \frac{π}{4} - \frac{y}{4}$

Step 8

Solve for $y$ .

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Step 8.1

Simplify the left side.

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Step 8.1.1

Simplify $\tan (4 (\frac{π}{2} - \frac{y}{4}) + y)$ .

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Step 8.1.1.1

Rewrite.

$0 + 0 + \tan (4 (\frac{π}{2} - \frac{y}{4}) + y) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.2

Simplify by adding zeros.

$\tan (4 (\frac{π}{2} - \frac{y}{4}) + y) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.3

Simplify each term.

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Step 8.1.1.3.1

Apply the distributive property.

$\tan (4 \frac{π}{2} + 4 (- \frac{y}{4}) + y) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.3.2

Cancel the common factor of $2$ .

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Step 8.1.1.3.2.1

Factor $2$ out of $4$ .

$\tan (2 (2) \frac{π}{2} + 4 (- \frac{y}{4}) + y) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.3.2.2

Cancel the common factor.

$\tan (2 \cdot 2 \frac{π}{2} + 4 (- \frac{y}{4}) + y) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.3.2.3

Rewrite the expression.

$\tan (2 π + 4 (- \frac{y}{4}) + y) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.3.3

Cancel the common factor of $4$ .

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Step 8.1.1.3.3.1

Move the leading negative in $- \frac{y}{4}$ into the numerator.

$\tan (2 π + 4 \frac{- y}{4} + y) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.3.3.2

Cancel the common factor.

$\tan (2 π + 4 \frac{- y}{4} + y) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.3.3.3

Rewrite the expression.

$\tan (2 π - y + y) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.4

Combine the opposite terms in $2 π - y + y$ .

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Step 8.1.1.4.1

Add $- y$ and $y$ .

$\tan (2 π + 0) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.4.2

Add $2 π$ and $0$ .

$\tan (2 π) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.5

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\tan (0) = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.1.1.6

The exact value of $\tan (0)$ is $0$ .

$0 = 4 (\frac{π}{2} - \frac{y}{4})$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $4 (\frac{π}{2} - \frac{y}{4})$ .

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Step 8.2.1.1

Apply the distributive property.

$0 = 4 \frac{π}{2} + 4 (- \frac{y}{4})$

Step 8.2.1.2

Cancel the common factor of $2$ .

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Step 8.2.1.2.1

Factor $2$ out of $4$ .

$0 = 2 (2) \frac{π}{2} + 4 (- \frac{y}{4})$

Step 8.2.1.2.2

Cancel the common factor.

$0 = 2 \cdot 2 \frac{π}{2} + 4 (- \frac{y}{4})$

Step 8.2.1.2.3

Rewrite the expression.

$0 = 2 π + 4 (- \frac{y}{4})$

Step 8.2.1.3

Cancel the common factor of $4$ .

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Step 8.2.1.3.1

Move the leading negative in $- \frac{y}{4}$ into the numerator.

$0 = 2 π + 4 \frac{- y}{4}$

Step 8.2.1.3.2

Cancel the common factor.

$0 = 2 π + 4 \frac{- y}{4}$

Step 8.2.1.3.3

Rewrite the expression.

$0 = 2 π - y$

Step 8.3

Rewrite the equation as $2 π - y = 0$ .

$2 π - y = 0$

Step 8.4

Subtract $2 π$ from both sides of the equation.

$- y = - 2 π$

Step 8.5

Divide each term in $- y = - 2 π$ by $- 1$ and simplify.

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Step 8.5.1

Divide each term in $- y = - 2 π$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 2 π}{- 1}$

Step 8.5.2

Simplify the left side.

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Step 8.5.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 2 π}{- 1}$

Step 8.5.2.2

Divide $y$ by $1$ .

$y = \frac{- 2 π}{- 1}$

Step 8.5.3

Simplify the right side.

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Step 8.5.3.1

Move the negative one from the denominator of $\frac{- 2 π}{- 1}$ .

$y = - 1 \cdot (- 2 π)$

Step 8.5.3.2

Rewrite $- 1 \cdot (- 2 π)$ as $- (- 2 π)$ .

$y = - (- 2 π)$

Step 8.5.3.3

Multiply $- 2$ by $- 1$ .

$y = 2 π$

Step 9

Solve for $y$ .

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Step 9.1

Simplify the left side.

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Step 9.1.1

Simplify $\tan (4 (\frac{π}{4} - \frac{y}{4}) + y)$ .

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Step 9.1.1.1

Rewrite.

$0 + 0 + \tan (4 (\frac{π}{4} - \frac{y}{4}) + y) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.2

Simplify by adding zeros.

$\tan (4 (\frac{π}{4} - \frac{y}{4}) + y) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.3

Simplify each term.

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Step 9.1.1.3.1

Apply the distributive property.

$\tan (4 \frac{π}{4} + 4 (- \frac{y}{4}) + y) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.3.2

Cancel the common factor of $4$ .

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Step 9.1.1.3.2.1

Cancel the common factor.

$\tan (4 \frac{π}{4} + 4 (- \frac{y}{4}) + y) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.3.2.2

Rewrite the expression.

$\tan (π + 4 (- \frac{y}{4}) + y) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.3.3

Cancel the common factor of $4$ .

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Step 9.1.1.3.3.1

Move the leading negative in $- \frac{y}{4}$ into the numerator.

$\tan (π + 4 \frac{- y}{4} + y) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.3.3.2

Cancel the common factor.

$\tan (π + 4 \frac{- y}{4} + y) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.3.3.3

Rewrite the expression.

$\tan (π - y + y) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.4

Combine the opposite terms in $π - y + y$ .

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Step 9.1.1.4.1

Add $- y$ and $y$ .

$\tan (π + 0) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.4.2

Add $π$ and $0$ .

$\tan (π) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$- \tan (0) = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.6

The exact value of $\tan (0)$ is $0$ .

$- 0 = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.1.1.7

Multiply $- 1$ by $0$ .

$0 = 4 (\frac{π}{4} - \frac{y}{4})$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $4 (\frac{π}{4} - \frac{y}{4})$ .

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Step 9.2.1.1

Apply the distributive property.

$0 = 4 \frac{π}{4} + 4 (- \frac{y}{4})$

Step 9.2.1.2

Cancel the common factor of $4$ .

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Step 9.2.1.2.1

Cancel the common factor.

$0 = 4 \frac{π}{4} + 4 (- \frac{y}{4})$

Step 9.2.1.2.2

Rewrite the expression.

$0 = π + 4 (- \frac{y}{4})$

Step 9.2.1.3

Cancel the common factor of $4$ .

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Step 9.2.1.3.1

Move the leading negative in $- \frac{y}{4}$ into the numerator.

$0 = π + 4 \frac{- y}{4}$

Step 9.2.1.3.2

Cancel the common factor.

$0 = π + 4 \frac{- y}{4}$

Step 9.2.1.3.3

Rewrite the expression.

$0 = π - y$

Step 9.3

Rewrite the equation as $π - y = 0$ .

$π - y = 0$

Step 9.4

Subtract $π$ from both sides of the equation.

$- y = - π$

Step 9.5

Divide each term in $- y = - π$ by $- 1$ and simplify.

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Step 9.5.1

Divide each term in $- y = - π$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- π}{- 1}$

Step 9.5.2

Simplify the left side.

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Step 9.5.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- π}{- 1}$

Step 9.5.2.2

Divide $y$ by $1$ .

$y = \frac{- π}{- 1}$

Step 9.5.3

Simplify the right side.

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Step 9.5.3.1

Dividing two negative values results in a positive value.

$y = \frac{π}{1}$

Step 9.5.3.2

Divide $π$ by $1$ .

$y = π$

Step 10

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

$(\frac{π}{2} - \frac{y}{4}, 2 π)$

$(\frac{π}{4} - \frac{y}{4}, π)$

Step 11