Calculus Examples

$y = \cot (12 π x - 3 x)$

Step 1

Write $y = \cot (12 π x - 3 x)$ as a function.

$f (x) = \cot (12 π x - 3 x)$

Step 2

Find the first derivative of the function.

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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 $g (x) = 12 π x - 3 x$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $12 π x - 3 x$ .

$\frac{d}{d u} [\cot (u)] \frac{d}{d x} [12 π x - 3 x]$

Step 2.1.2

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

$- \csc^{2} (u) \frac{d}{d x} [12 π x - 3 x]$

Step 2.1.3

Replace all occurrences of $u$ with $12 π x - 3 x$ .

$- \csc^{2} (12 π x - 3 x) \frac{d}{d x} [12 π x - 3 x]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $12 π x - 3 x$ with respect to $x$ is $\frac{d}{d x} [12 π x] + \frac{d}{d x} [- 3 x]$ .

$- \csc^{2} (12 π x - 3 x) (\frac{d}{d x} [12 π x] + \frac{d}{d x} [- 3 x])$

Step 2.2.2

Since $12 π$ is constant with respect to $x$ , the derivative of $12 π x$ with respect to $x$ is $12 π \frac{d}{d x} [x]$ .

$- \csc^{2} (12 π x - 3 x) (12 π \frac{d}{d x} [x] + \frac{d}{d x} [- 3 x])$

Step 2.2.3

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

$- \csc^{2} (12 π x - 3 x) (12 π \cdot 1 + \frac{d}{d x} [- 3 x])$

Step 2.2.4

Multiply $12$ by $1$ .

$- \csc^{2} (12 π x - 3 x) (12 π + \frac{d}{d x} [- 3 x])$

Step 2.2.5

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

$- \csc^{2} (12 π x - 3 x) (12 π - 3 \frac{d}{d x} [x])$

Step 2.2.6

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

$- \csc^{2} (12 π x - 3 x) (12 π - 3 \cdot 1)$

Step 2.2.7

Multiply $- 3$ by $1$ .

$- \csc^{2} (12 π x - 3 x) (12 π - 3)$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $- (12 π - 3)$ is constant with respect to $x$ , the derivative of $- \csc^{2} (12 π x - 3 x) (12 π - 3)$ with respect to $x$ is $- (12 π - 3) \frac{d}{d x} [\csc^{2} (12 π x - 3 x)]$ .

$f'' (x) = - (12 π - 3) \frac{d}{d x} \csc^{2} (12 π x - 3 x)$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \csc (12 π x - 3 x)$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $\csc (12 π x - 3 x)$ .

$f'' (x) = - (12 π - 3) (\frac{d}{d u (1)} ({u_{1}}^{2}) \frac{d}{d x} (\csc (12 π x - 3 x)))$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$f'' (x) = - (12 π - 3) (2 u_{1} \frac{d}{d x} (\csc (12 π x - 3 x)))$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $\csc (12 π x - 3 x)$ .

$f'' (x) = - (12 π - 3) (2 \csc (12 π x - 3 x) \frac{d}{d x} (\csc (12 π x - 3 x)))$

Step 3.3

Multiply $2$ by $- 1$ .

$f'' (x) = - 2 (12 π - 3) (\csc (12 π x - 3 x) \frac{d}{d x} (\csc (12 π x - 3 x)))$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \csc (x)$ and $g (x) = 12 π x - 3 x$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $12 π x - 3 x$ .

$f'' (x) = - 2 (12 π - 3) \csc (12 π x - 3 x) (\frac{d}{d u (2)} (\csc (u_{2})) \frac{d}{d x} (12 π x - 3 x))$

Step 3.4.2

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

$f'' (x) = - 2 (12 π - 3) \csc (12 π x - 3 x) (- \csc (u_{2}) \cot (u_{2}) \frac{d}{d x} (12 π x - 3 x))$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $12 π x - 3 x$ .

$f'' (x) = - 2 (12 π - 3) \csc (12 π x - 3 x) (- \csc (12 π x - 3 x) \cot (12 π x - 3 x) \frac{d}{d x} (12 π x - 3 x))$

Step 3.5

Multiply $- 1$ by $- 2$ .

$f'' (x) = 2 (12 π - 3) \csc (12 π x - 3 x) (\csc (12 π x - 3 x) \cot (12 π x - 3 x) \frac{d}{d x} (12 π x - 3 x))$

Step 3.6

Raise $\csc (12 π x - 3 x)$ to the power of $1$ .

$f'' (x) = 2 (12 π - 3) (\csc (12 π x - 3 x) \csc (12 π x - 3 x)) (\cot (12 π x - 3 x) \frac{d}{d x} (12 π x - 3 x))$

Step 3.7

Raise $\csc (12 π x - 3 x)$ to the power of $1$ .

$f'' (x) = 2 (12 π - 3) (\csc (12 π x - 3 x) \csc (12 π x - 3 x)) (\cot (12 π x - 3 x) \frac{d}{d x} (12 π x - 3 x))$

Step 3.8

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

$f'' (x) = 2 (12 π - 3) {\csc (12 π x - 3 x)}^{1 + 1} (\cot (12 π x - 3 x) \frac{d}{d x} (12 π x - 3 x))$

Step 3.9

Add $1$ and $1$ .

$f'' (x) = 2 (12 π - 3) \csc^{2} (12 π x - 3 x) (\cot (12 π x - 3 x) \frac{d}{d x} (12 π x - 3 x))$

Step 3.10

By the Sum Rule, the derivative of $12 π x - 3 x$ with respect to $x$ is $\frac{d}{d x} [12 π x] + \frac{d}{d x} [- 3 x]$ .

$f'' (x) = 2 (12 π - 3) \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x) (\frac{d}{d x} (12 π x) + \frac{d}{d x} (- 3 x))$

Step 3.11

Since $12 π$ is constant with respect to $x$ , the derivative of $12 π x$ with respect to $x$ is $12 π \frac{d}{d x} [x]$ .

$f'' (x) = 2 (12 π - 3) \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x) (12 π \frac{d}{d x} (x) + \frac{d}{d x} (- 3 x))$

Step 3.12

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

$f'' (x) = 2 (12 π - 3) \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x) (12 π \cdot 1 + \frac{d}{d x} (- 3 x))$

Step 3.13

Multiply $12$ by $1$ .

$f'' (x) = 2 (12 π - 3) \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x) (12 π + \frac{d}{d x} (- 3 x))$

Step 3.14

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

$f'' (x) = 2 (12 π - 3) \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x) (12 π - 3 \frac{d}{d x} x)$

Step 3.15

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

$f'' (x) = 2 (12 π - 3) \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x) (12 π - 3 \cdot 1)$

Step 3.16

Multiply $- 3$ by $1$ .

$f'' (x) = 2 (12 π - 3) \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x) (12 π - 3)$

Step 3.17

Raise $12 π - 3$ to the power of $1$ .

$f'' (x) = 2 ((12 π - 3) (12 π - 3)) \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x)$

Step 3.18

Raise $12 π - 3$ to the power of $1$ .

$f'' (x) = 2 ((12 π - 3) (12 π - 3)) \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x)$

Step 3.19

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

$f'' (x) = 2 {(12 π - 3)}^{1 + 1} \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x)$

Step 3.20

Add $1$ and $1$ .

$f'' (x) = 2 {(12 π - 3)}^{2} \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x)$

Step 3.21

Reorder the factors of $2 {(12 π - 3)}^{2} \csc^{2} (12 π x - 3 x) \cot (12 π x - 3 x)$ .

$f'' (x) = 2 \csc^{2} (12 π x - 3 x) {(12 π - 3)}^{2} \cot (12 π x - 3 x)$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \csc^{2} (12 π x - 3 x) (12 π - 3) = 0$

Step 5

Divide each term in $- \csc^{2} (12 π x - 3 x) (12 π - 3) = 0$ by $- 12 π + 3$ and simplify.

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Step 5.1

Divide each term in $- \csc^{2} (12 π x - 3 x) (12 π - 3) = 0$ by $- 12 π + 3$ .

$\frac{- \csc^{2} (12 π x - 3 x) (12 π - 3)}{- 12 π + 3} = \frac{0}{- 12 π + 3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $12 π - 3$ and $- 12 π + 3$ .

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Step 5.2.1.1

Factor $3$ out of $- \csc^{2} (12 π x - 3 x) (12 π - 3)$ .

$\frac{3 (- \csc^{2} (12 π x - 3 x) (4 π - 1))}{- 12 π + 3} = \frac{0}{- 12 π + 3}$

Step 5.2.1.2

Cancel the common factors.

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Step 5.2.1.2.1

Factor $3$ out of $- 12 π$ .

$\frac{3 (- \csc^{2} (12 π x - 3 x) (4 π - 1))}{3 (- 4 π) + 3} = \frac{0}{- 12 π + 3}$

Step 5.2.1.2.2

Factor $3$ out of $3$ .

$\frac{3 (- \csc^{2} (12 π x - 3 x) (4 π - 1))}{3 (- 4 π) + 3 (1)} = \frac{0}{- 12 π + 3}$

Step 5.2.1.2.3

Factor $3$ out of $3 (- 4 π) + 3 (1)$ .

$\frac{3 (- \csc^{2} (12 π x - 3 x) (4 π - 1))}{3 (- 4 π + 1)} = \frac{0}{- 12 π + 3}$

Step 5.2.1.2.4

Cancel the common factor.

$\frac{3 (- \csc^{2} (12 π x - 3 x) (4 π - 1))}{3 (- 4 π + 1)} = \frac{0}{- 12 π + 3}$

Step 5.2.1.2.5

Rewrite the expression.

$\frac{- \csc^{2} (12 π x - 3 x) (4 π - 1)}{- 4 π + 1} = \frac{0}{- 12 π + 3}$

Step 5.2.2

Cancel the common factor of $4 π - 1$ and $- 4 π + 1$ .

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Step 5.2.2.1

Factor $- 1$ out of $4 π$ .

$\frac{- \csc^{2} (12 π x - 3 x) (- (- 4 π) - 1)}{- 4 π + 1} = \frac{0}{- 12 π + 3}$

Step 5.2.2.2

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

$\frac{- \csc^{2} (12 π x - 3 x) (- (- 4 π) - 1 (1))}{- 4 π + 1} = \frac{0}{- 12 π + 3}$

Step 5.2.2.3

Factor $- 1$ out of $- (- 4 π) - 1 (1)$ .

$\frac{- \csc^{2} (12 π x - 3 x) (- (- 4 π + 1))}{- 4 π + 1} = \frac{0}{- 12 π + 3}$

Step 5.2.2.4

Rewrite $- (- 4 π + 1)$ as $- 1 (- 4 π + 1)$ .

$\frac{- \csc^{2} (12 π x - 3 x) (- 1 (- 4 π + 1))}{- 4 π + 1} = \frac{0}{- 12 π + 3}$

Step 5.2.2.5

Cancel the common factor.

$\frac{- \csc^{2} (12 π x - 3 x) (- 1 (- 4 π + 1))}{- 4 π + 1} = \frac{0}{- 12 π + 3}$

Step 5.2.2.6

Divide $- \csc^{2} (12 π x - 3 x) \cdot - 1$ by $1$ .

$- \csc^{2} (12 π x - 3 x) \cdot - 1 = \frac{0}{- 12 π + 3}$

Step 5.2.3

Multiply $- \csc^{2} (12 π x - 3 x) \cdot - 1$ .

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Step 5.2.3.1

Multiply $- 1$ by $- 1$ .

$1 \csc^{2} (12 π x - 3 x) = \frac{0}{- 12 π + 3}$

Step 5.2.3.2

Multiply $\csc^{2} (12 π x - 3 x)$ by $1$ .

$\csc^{2} (12 π x - 3 x) = \frac{0}{- 12 π + 3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $0$ and $- 12 π + 3$ .

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Step 5.3.1.1

Factor $3$ out of $0$ .

$\csc^{2} (12 π x - 3 x) = \frac{3 (0)}{- 12 π + 3}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $3$ out of $- 12 π$ .

$\csc^{2} (12 π x - 3 x) = \frac{3 (0)}{3 (- 4 π) + 3}$

Step 5.3.1.2.2

Factor $3$ out of $3$ .

$\csc^{2} (12 π x - 3 x) = \frac{3 \cdot 0}{3 (- 4 π) + 3 \cdot 1}$

Step 5.3.1.2.3

Factor $3$ out of $3 (- 4 π) + 3 (1)$ .

$\csc^{2} (12 π x - 3 x) = \frac{3 (0)}{3 (- 4 π + 1)}$

Step 5.3.1.2.4

Cancel the common factor.

$\csc^{2} (12 π x - 3 x) = \frac{3 \cdot 0}{3 (- 4 π + 1)}$

Step 5.3.1.2.5

Rewrite the expression.

$\csc^{2} (12 π x - 3 x) = \frac{0}{- 4 π + 1}$

Step 5.3.2

Divide $0$ by $- 4 π + 1$ .

$\csc^{2} (12 π x - 3 x) = 0$

Step 6

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

$\csc (12 π x - 3 x) = \pm \sqrt{0}$

Step 7

Simplify $\pm \sqrt{0}$ .

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Step 7.1

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

$\csc (12 π x - 3 x) = \pm \sqrt{0^{2}}$

Step 7.2

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

$\csc (12 π x - 3 x) = \pm 0$

Step 7.3

Plus or minus $0$ is $0$ .

$\csc (12 π x - 3 x) = 0$

Step 8

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 9

Evaluate the second derivative at $x =$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$2 \csc^{2} (12 π - 3) {(12 π - 3)}^{2} \cot (12 π - 3)$

Step 10

Evaluate the second derivative.

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Step 10.1

Evaluate $\csc (12 π - 3)$ .

$2 {(- 7.08616739)}^{2} {(12 π - 3)}^{2} \cot (12 π - 3)$

Step 10.2

Simplify the expression.

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Step 10.2.1

Raise $- 7.08616739$ to the power of $2$ .

$2 \cdot 50.21376836 {(12 π - 3)}^{2} \cot (12 π - 3)$

Step 10.2.2

Multiply $2$ by $50.21376836$ .

$100.42753672 {(12 π - 3)}^{2} \cot (12 π - 3)$

Step 10.2.3

Rewrite ${(12 π - 3)}^{2}$ as $(12 π - 3) (12 π - 3)$ .

$100.42753672 ((12 π - 3) (12 π - 3)) \cot (12 π - 3)$

Step 10.3

Expand $(12 π - 3) (12 π - 3)$ using the FOIL Method.

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Step 10.3.1

Apply the distributive property.

$100.42753672 (12 π (12 π - 3) - 3 (12 π - 3)) \cot (12 π - 3)$

Step 10.3.2

Apply the distributive property.

$100.42753672 (12 π (12 π) + 12 π \cdot - 3 - 3 (12 π - 3)) \cot (12 π - 3)$

Step 10.3.3

Apply the distributive property.

$100.42753672 (12 π (12 π) + 12 π \cdot - 3 - 3 (12 π) - 3 \cdot - 3) \cot (12 π - 3)$

Step 10.4

Simplify and combine like terms.

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Step 10.4.1

Simplify each term.

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Step 10.4.1.1

Multiply $12 π (12 π)$ .

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Step 10.4.1.1.1

Multiply $12$ by $12$ .

$100.42753672 (144 π π + 12 π \cdot - 3 - 3 (12 π) - 3 \cdot - 3) \cot (12 π - 3)$

Step 10.4.1.1.2

Raise $π$ to the power of $1$ .

$100.42753672 (144 (π^{1} π) + 12 π \cdot - 3 - 3 (12 π) - 3 \cdot - 3) \cot (12 π - 3)$

Step 10.4.1.1.3

Raise $π$ to the power of $1$ .

$100.42753672 (144 (π^{1} π^{1}) + 12 π \cdot - 3 - 3 (12 π) - 3 \cdot - 3) \cot (12 π - 3)$

Step 10.4.1.1.4

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

$100.42753672 (144 π^{1 + 1} + 12 π \cdot - 3 - 3 (12 π) - 3 \cdot - 3) \cot (12 π - 3)$

Step 10.4.1.1.5

Add $1$ and $1$ .

$100.42753672 (144 π^{2} + 12 π \cdot - 3 - 3 (12 π) - 3 \cdot - 3) \cot (12 π - 3)$

Step 10.4.1.2

Multiply $- 3$ by $12$ .

$100.42753672 (144 π^{2} - 36 π - 3 (12 π) - 3 \cdot - 3) \cot (12 π - 3)$

Step 10.4.1.3

Multiply $12$ by $- 3$ .

$100.42753672 (144 π^{2} - 36 π - 36 π - 3 \cdot - 3) \cot (12 π - 3)$

Step 10.4.1.4

Multiply $- 3$ by $- 3$ .

$100.42753672 (144 π^{2} - 36 π - 36 π + 9) \cot (12 π - 3)$

Step 10.4.2

Subtract $36 π$ from $- 36 π$ .

$100.42753672 (144 π^{2} - 72 π + 9) \cot (12 π - 3)$

Step 10.5

Multiply $100.42753672$ by $144 π^{2} - 72 π + 9$ .

$120917.6026078 \cot (12 π - 3)$

Step 10.6

Evaluate $\cot (12 π - 3)$ .

$120917.6026078 \cdot 7.01525255$

Step 10.7

Multiply $120917.6026078$ by $7.01525255$ .

$848267.52020773$

Step 11

$x =$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x =$ is a local minimum

Step 12

These are the local extrema for $f (x) = \cot (12 π x - 3 x)$ .

$(, i s a (l o c a l) (m i n i m u m))$ is a local minima

Step 13