Calculus Examples

$x^{6} = \cot (y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{6}) = \frac{d}{d x} (\cot (y))$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$6 x^{5}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 4

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

$6 x^{5} = - \csc^{2} (y) y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- \csc^{2} (y) y' = 6 x^{5}$ .

$- \csc^{2} (y) y' = 6 x^{5}$

Step 5.2

Divide each term in $- \csc^{2} (y) y' = 6 x^{5}$ by $- \csc^{2} (y)$ and simplify.

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Step 5.2.1

Divide each term in $- \csc^{2} (y) y' = 6 x^{5}$ by $- \csc^{2} (y)$ .

$\frac{- \csc^{2} (y) y'}{- \csc^{2} (y)} = \frac{6 x^{5}}{- \csc^{2} (y)}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\csc^{2} (y) y'}{\csc^{2} (y)} = \frac{6 x^{5}}{- \csc^{2} (y)}$

Step 5.2.2.2

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

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{\csc^{2} (y) y'}{\csc^{2} (y)} = \frac{6 x^{5}}{- \csc^{2} (y)}$

Step 5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{6 x^{5}}{- \csc^{2} (y)}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$y' = - \frac{6 x^{5}}{\csc^{2} (y)}$

Step 6

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

$\frac{d y}{d x} = - \frac{6 x^{5}}{\csc^{2} (y)}$

Step 7

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

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Step 7.1

Set the numerator equal to zero.

$6 x^{5} = 0$

Step 7.2

Solve the equation for $x$ .

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Step 7.2.1

Divide each term in $6 x^{5} = 0$ by $6$ and simplify.

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Step 7.2.1.1

Divide each term in $6 x^{5} = 0$ by $6$ .

$\frac{6 x^{5}}{6} = \frac{0}{6}$

Step 7.2.1.2

Simplify the left side.

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Step 7.2.1.2.1

Cancel the common factor of $6$ .

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Step 7.2.1.2.1.1

Cancel the common factor.

$\frac{6 x^{5}}{6} = \frac{0}{6}$

Step 7.2.1.2.1.2

Divide $x^{5}$ by $1$ .

$x^{5} = \frac{0}{6}$

Step 7.2.1.3

Simplify the right side.

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Step 7.2.1.3.1

Divide $0$ by $6$ .

$x^{5} = 0$

Step 7.2.2

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

$x = \sqrt[5]{0}$

Step 7.2.3

Simplify $\sqrt[5]{0}$ .

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Step 7.2.3.1

Rewrite $0$ as $0^{5}$ .

$x = \sqrt[5]{0^{5}}$

Step 7.2.3.2

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

$x = 0$

Step 8

Solve for $y$ .

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Step 8.1

Rewrite the equation as $\cot (y) = 0^{6}$ .

$\cot (y) = 0^{6}$

Step 8.2

Raising $0$ to any positive power yields $0$ .

$\cot (y) = 0$

Step 8.3

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

$y = arccot (0)$

Step 8.4

Simplify the right side.

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Step 8.4.1

The exact value of $arccot (0)$ is $\frac{π}{2}$ .

$y = \frac{π}{2}$

Step 8.5

The cotangent 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.

$y = π + \frac{π}{2}$

Step 8.6

Simplify $π + \frac{π}{2}$ .

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Step 8.6.1

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

$y = π \cdot \frac{2}{2} + \frac{π}{2}$

Step 8.6.2

Combine fractions.

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Step 8.6.2.1

Combine $π$ and $\frac{2}{2}$ .

$y = \frac{π \cdot 2}{2} + \frac{π}{2}$

Step 8.6.2.2

Combine the numerators over the common denominator.

$y = \frac{π \cdot 2 + π}{2}$

Step 8.6.3

Simplify the numerator.

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Step 8.6.3.1

Move $2$ to the left of $π$ .

$y = \frac{2 \cdot π + π}{2}$

Step 8.6.3.2

Add $2 π$ and $π$ .

$y = \frac{3 π}{2}$

Step 8.7

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

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Step 8.7.1

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

$\frac{π}{| b |}$

Step 8.7.2

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

$\frac{π}{| 1 |}$

Step 8.7.3

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

$\frac{π}{1}$

Step 8.7.4

Divide $π$ by $1$ .

$π$

Step 8.8

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

$y = \frac{π}{2} + π n, \frac{3 π}{2} + π n$ , for any integer $n$

Step 8.9

Consolidate the answers.

$y = \frac{π}{2} + π n$ , for any integer $n$

Step 9

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

$(0, \frac{π}{2} + π n)$

Step 10