Calculus Examples

$\cot (y) = x - y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\cot (y)) = \frac{d}{d x} (x - y)$

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 $f (x) = \cot (x)$ and $g (x) = y$ .

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Step 2.1.1

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

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

Step 2.1.2

The derivative of $\cot (u)$ with respect to $u$ is $- \csc^{2} (u)$ .

$- \csc^{2} (u) \frac{d}{d x} [y]$

Step 2.1.3

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

$- \csc^{2} (y) \frac{d}{d x} [y]$

Step 2.2

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

$- \csc^{2} (y) y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

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

$\frac{d}{d x} [x] + \frac{d}{d x} [- y]$

Step 3.1.2

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

$1 + \frac{d}{d x} [- y]$

Step 3.2

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

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Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- y$ with respect to $x$ is $- \frac{d}{d x} [y]$ .

$1 - \frac{d}{d x} [y]$

Step 3.2.2

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

$1 - y'$

Step 4

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

$- \csc^{2} (y) y' = 1 - y'$

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 $- \csc^{2} (y) y'$ .

$- y' \csc^{2} (y) = 1 - y'$

Step 5.2

Move all terms containing $y'$ to the left side of the equation.

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Step 5.2.1

Add $y'$ to both sides of the equation.

$- y' \csc^{2} (y) + y' = 1$

Step 5.2.2

Reorder $- y' \csc^{2} (y)$ and $y'$ .

$y' - y' \csc^{2} (y) = 1$

Step 5.2.3

Rewrite $y'$ as $1 y'$ .

$1 y' - y' \csc^{2} (y) = 1$

Step 5.2.4

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

$- y' \cdot - 1 - y' \csc^{2} (y) = 1$

Step 5.2.5

Factor $- y'$ out of $- y' \csc^{2} (y)$ .

$- y' \cdot - 1 - y' \csc^{2} (y) = 1$

Step 5.2.6

Factor $- y'$ out of $- y' \cdot - 1 - y' \csc^{2} (y)$ .

$- y' (- 1 + \csc^{2} (y)) = 1$

Step 5.2.7

Rearrange terms.

$- y' (\csc^{2} (y) - 1) = 1$

Step 5.2.8

Apply pythagorean identity.

$- y' \cot^{2} (y) = 1$

Step 5.3

Divide each term in $- y' \cot^{2} (y) = 1$ by $- \cot^{2} (y)$ and simplify.

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Step 5.3.1

Divide each term in $- y' \cot^{2} (y) = 1$ by $- \cot^{2} (y)$ .

$\frac{- y' \cot^{2} (y)}{- \cot^{2} (y)} = \frac{1}{- \cot^{2} (y)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Dividing two negative values results in a positive value.

$\frac{y' \cot^{2} (y)}{\cot^{2} (y)} = \frac{1}{- \cot^{2} (y)}$

Step 5.3.2.2

Cancel the common factor of $\cot^{2} (y)$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' \cot^{2} (y)}{\cot^{2} (y)} = \frac{1}{- \cot^{2} (y)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{1}{- \cot^{2} (y)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 5.3.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$y' = \frac{- 1 (- 1)}{- \cot^{2} (y)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{1}{\cot^{2} (y)}$

Step 5.3.3.2

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

$y' = - \frac{1^{2}}{\cot^{2} (y)}$

Step 5.3.3.3

Rewrite $\frac{1^{2}}{\cot^{2} (y)}$ as ${(\frac{1}{\cot (y)})}^{2}$ .

$y' = - {(\frac{1}{\cot (y)})}^{2}$

Step 5.3.3.4

Rewrite $\cot (y)$ in terms of sines and cosines.

$y' = - {(\frac{1}{\frac{\cos (y)}{\sin (y)}})}^{2}$

Step 5.3.3.5

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (y)}{\sin (y)}$ .

$y' = - {(\frac{\sin (y)}{\cos (y)})}^{2}$

Step 5.3.3.6

Convert from $\frac{\sin (y)}{\cos (y)}$ to $\tan (y)$ .

$y' = - \tan^{2} (y)$

Step 6

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

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

Step 7

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

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Step 7.1

Divide each term in $- \tan^{2} (y) = 0$ by $- 1$ and simplify.

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Step 7.1.1

Divide each term in $- \tan^{2} (y) = 0$ by $- 1$ .

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

Step 7.1.2

Simplify the left side.

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Step 7.1.2.1

Dividing two negative values results in a positive value.

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

Step 7.1.2.2

Divide $\tan^{2} (y)$ by $1$ .

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

Step 7.1.3

Simplify the right side.

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Step 7.1.3.1

Divide $0$ by $- 1$ .

$\tan^{2} (y) = 0$

Step 7.2

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

$\tan (y) = \pm \sqrt{0}$

Step 7.3

Simplify $\pm \sqrt{0}$ .

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Step 7.3.1

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

$\tan (y) = \pm \sqrt{0^{2}}$

Step 7.3.2

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

$\tan (y) = \pm 0$

Step 7.3.3

Plus or minus $0$ is $0$ .

$\tan (y) = 0$

Step 7.4

Take the inverse tangent of both sides of the equation to extract $y$ from inside the tangent.

$y = \arctan (0)$

Step 7.5

Simplify the right side.

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Step 7.5.1

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

$y = 0$

Step 7.6

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$y = π + 0$

Step 7.7

Add $π$ and $0$ .

$y = π$

Step 7.8

Find the period of $\tan (y)$ .

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Step 7.8.1

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

$\frac{π}{| b |}$

Step 7.8.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 7.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 7.8.4

Divide $π$ by $1$ .

$π$

Step 7.9

The period of the $\tan (y)$ function is $π$ so values will repeat every $π$ radians in both directions.

$y = π n, π + π n$ , for any integer $n$

Step 7.10

Consolidate the answers.

$y = π n$ , for any integer $n$

Step 8

Simplify the right side.

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Step 8.1

Remove parentheses.

$\cot (y) = π n - y$

Step 9

Solve for $x$ when $y$ is $π n - y$ .

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Step 9.1

Move all terms containing $n$ to the left side of the equation.

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Step 9.1.1

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

$π n - y - π n = 0$

Step 9.1.2

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

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Step 9.1.2.1

Subtract $π n$ from $π n$ .

$- y + 0 = 0$

Step 9.1.2.2

Add $- y$ and $0$ .

$- y = 0$

Step 9.2

Divide each term in $- y = 0$ by $- 1$ and simplify.

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Step 9.2.1

Divide each term in $- y = 0$ by $- 1$ .

$\frac{- y}{- 1} = \frac{0}{- 1}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{0}{- 1}$

Step 9.2.2.2

Divide $y$ by $1$ .

$y = \frac{0}{- 1}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Divide $0$ by $- 1$ .

$y = 0$

Step 10

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

$(0, π n - y)$

Step 11