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The derivative of the function/ log(cot(x/6)^(3))

Derivative of log(cot(x/6)^(3))

The solution

You have entered [src]

   /   3/x\\
log|cot |-||
   \    \6//

$$\log{\left(\cot^{3}{\left(\frac{x}{6} \right)} \right)}$$

log(cot(x/6)^3)

Detail solution

Let $u = \cot^{3}{\left(\frac{x}{6} \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot^{3}{\left(\frac{x}{6} \right)}$ :
1. Let $u = \cot{\left(\frac{x}{6} \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(\frac{x}{6} \right)}$ :
  1. There are multiple ways to do this derivative.
    Method #1
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(\frac{x}{6} \right)} = \frac{1}{\tan{\left(\frac{x}{6} \right)}}$
    2. Let $u = \tan{\left(\frac{x}{6} \right)}$ .
    3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\frac{x}{6} \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(\frac{x}{6} \right)} = \frac{\sin{\left(\frac{x}{6} \right)}}{\cos{\left(\frac{x}{6} \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(\frac{x}{6} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{x}{6} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = \frac{x}{6}$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{6}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $\frac{1}{6}$
      
      The result of the chain rule is:
      
      $\frac{\cos{\left(\frac{x}{6} \right)}}{6}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \frac{x}{6}$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{6}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $\frac{1}{6}$
      
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\frac{x}{6} \right)}}{6}$
      
      Now plug in to the quotient rule:
      
      $\frac{\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}}{\cos^{2}{\left(\frac{x}{6} \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}}{\cos^{2}{\left(\frac{x}{6} \right)} \tan^{2}{\left(\frac{x}{6} \right)}}$
    Method #2
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(\frac{x}{6} \right)} = \frac{\cos{\left(\frac{x}{6} \right)}}{\sin{\left(\frac{x}{6} \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(\frac{x}{6} \right)}$ and $g{\left(x \right)} = \sin{\left(\frac{x}{6} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = \frac{x}{6}$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{6}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $\frac{1}{6}$
      
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\frac{x}{6} \right)}}{6}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \frac{x}{6}$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{6}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $\frac{1}{6}$
      
      The result of the chain rule is:
      
      $\frac{\cos{\left(\frac{x}{6} \right)}}{6}$
      
      Now plug in to the quotient rule:
      
      $\frac{- \frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} - \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}}{\sin^{2}{\left(\frac{x}{6} \right)}}$
  The result of the chain rule is:
  
  $- \frac{3 \left(\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}\right) \cot^{2}{\left(\frac{x}{6} \right)}}{\cos^{2}{\left(\frac{x}{6} \right)} \tan^{2}{\left(\frac{x}{6} \right)}}$
The result of the chain rule is:

$- \frac{3 \left(\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}\right)}{\cos^{2}{\left(\frac{x}{6} \right)} \tan^{2}{\left(\frac{x}{6} \right)} \cot{\left(\frac{x}{6} \right)}}$
Now simplify:

$- \frac{1}{\sin{\left(\frac{x}{3} \right)}}$

The answer is:

$- \frac{1}{\sin{\left(\frac{x}{3} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2/x\
      cot |-|
  1       \6/
- - - -------
  2      2   
-------------
       /x\   
    cot|-|   
       \6/

$$\frac{- \frac{\cot^{2}{\left(\frac{x}{6} \right)}}{2} - \frac{1}{2}}{\cot{\left(\frac{x}{6} \right)}}$$

Simplify

The second derivative [src]

                             2
                /       2/x\\ 
                |1 + cot |-|| 
         2/x\   \        \6// 
2 + 2*cot |-| - --------------
          \6/         2/x\    
                   cot |-|    
                       \6/    
------------------------------
              12

$$\frac{- \frac{\left(\cot^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}}{\cot^{2}{\left(\frac{x}{6} \right)}} + 2 \cot^{2}{\left(\frac{x}{6} \right)} + 2}{12}$$

Simplify

The third derivative [src]

              /                          2                  \
              |             /       2/x\\      /       2/x\\|
              |             |1 + cot |-||    2*|1 + cot |-|||
/       2/x\\ |       /x\   \        \6//      \        \6//|
|1 + cot |-||*|- 2*cot|-| - -------------- + ---------------|
\        \6// |       \6/         3/x\               /x\    |
              |                cot |-|            cot|-|    |
              \                    \6/               \6/    /
-------------------------------------------------------------
                              36

$$\frac{\left(\cot^{2}{\left(\frac{x}{6} \right)} + 1\right) \left(- \frac{\left(\cot^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}}{\cot^{3}{\left(\frac{x}{6} \right)}} + \frac{2 \left(\cot^{2}{\left(\frac{x}{6} \right)} + 1\right)}{\cot{\left(\frac{x}{6} \right)}} - 2 \cot{\left(\frac{x}{6} \right)}\right)}{36}$$

Simplify

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Identical expressions

Derivative of log(cot(x/6)^(3))

The graph:

Piecewise:

The solution

Method #1

Method #2