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Identical expressions
six *cot((two *x-pi)/ five)/ five
6 multiply by cotangent of ((2 multiply by x minus Pi ) divide by 5) divide by 5
six multiply by cotangent of ((two multiply by x minus Pi ) divide by five) divide by five
6cot((2x-pi)/5)/5
6cot2x-pi/5/5
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Similar expressions
6*cot((2*x+pi)/5)/5

The derivative of the function/ 6*cot((2*x-pi)/5)/5

Derivative of 6cot((2x-pi)/5)/5

The solution

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     /2*x - pi\
6*cot|--------|
     \   5    /
---------------
       5

$$\frac{6 \cot{\left(\frac{2 x - \pi}{5} \right)}}{5}$$

(6*cot((2*x - pi)/5))/5

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. There are multiple ways to do this derivative.
    Method #1
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(\frac{2 x - \pi}{5} \right)} = \frac{1}{\tan{\left(\frac{2 x - \pi}{5} \right)}}$
    2. Let $u = \tan{\left(\frac{2 x - \pi}{5} \right)}$ .
    3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\frac{2 x - \pi}{5} \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(\frac{2 x - \pi}{5} \right)} = \frac{\sin{\left(\frac{2 x - \pi}{5} \right)}}{\cos{\left(\frac{2 x - \pi}{5} \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(\frac{2 x - \pi}{5} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{2 x - \pi}{5} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = \frac{2 x - \pi}{5}$ .
      
      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 $\frac{d}{d x} \frac{2 x - \pi}{5}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Differentiate $2 x - \pi$ term by term:
      
      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: $2$
      
      The derivative of the constant $- \pi$ is zero.
      
      The result is: $2$
      
      So, the result is: $\frac{2}{5}$
      
      The result of the chain rule is:
      
      $\frac{2 \cos{\left(\frac{2 x - \pi}{5} \right)}}{5}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \frac{2 x - \pi}{5}$ .
      
      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 $\frac{d}{d x} \frac{2 x - \pi}{5}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Differentiate $2 x - \pi$ term by term:
      
      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: $2$
      
      The derivative of the constant $- \pi$ is zero.
      
      The result is: $2$
      
      So, the result is: $\frac{2}{5}$
      
      The result of the chain rule is:
      
      $- \frac{2 \sin{\left(\frac{2 x - \pi}{5} \right)}}{5}$
      
      Now plug in to the quotient rule:
      
      $\frac{\frac{2 \sin^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5} + \frac{2 \cos^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5}}{\cos^{2}{\left(\frac{2 x - \pi}{5} \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{\frac{2 \sin^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5} + \frac{2 \cos^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5}}{\cos^{2}{\left(\frac{2 x - \pi}{5} \right)} \tan^{2}{\left(\frac{2 x - \pi}{5} \right)}}$
    Method #2
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(\frac{2 x - \pi}{5} \right)} = \frac{\cos{\left(\frac{2 x - \pi}{5} \right)}}{\sin{\left(\frac{2 x - \pi}{5} \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(\frac{2 x - \pi}{5} \right)}$ and $g{\left(x \right)} = \sin{\left(\frac{2 x - \pi}{5} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = \frac{2 x - \pi}{5}$ .
      
      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 $\frac{d}{d x} \frac{2 x - \pi}{5}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Differentiate $2 x - \pi$ term by term:
      
      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: $2$
      
      The derivative of the constant $- \pi$ is zero.
      
      The result is: $2$
      
      So, the result is: $\frac{2}{5}$
      
      The result of the chain rule is:
      
      $- \frac{2 \sin{\left(\frac{2 x - \pi}{5} \right)}}{5}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \frac{2 x - \pi}{5}$ .
      
      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 $\frac{d}{d x} \frac{2 x - \pi}{5}$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Differentiate $2 x - \pi$ term by term:
      
      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: $2$
      
      The derivative of the constant $- \pi$ is zero.
      
      The result is: $2$
      
      So, the result is: $\frac{2}{5}$
      
      The result of the chain rule is:
      
      $\frac{2 \cos{\left(\frac{2 x - \pi}{5} \right)}}{5}$
      
      Now plug in to the quotient rule:
      
      $\frac{- \frac{2 \sin^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5} - \frac{2 \cos^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5}}{\sin^{2}{\left(\frac{2 x - \pi}{5} \right)}}$
  So, the result is: $- \frac{6 \left(\frac{2 \sin^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5} + \frac{2 \cos^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5}\right)}{\cos^{2}{\left(\frac{2 x - \pi}{5} \right)} \tan^{2}{\left(\frac{2 x - \pi}{5} \right)}}$
So, the result is: $- \frac{6 \left(\frac{2 \sin^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5} + \frac{2 \cos^{2}{\left(\frac{2 x - \pi}{5} \right)}}{5}\right)}{5 \cos^{2}{\left(\frac{2 x - \pi}{5} \right)} \tan^{2}{\left(\frac{2 x - \pi}{5} \right)}}$
Now simplify:

$- \frac{12}{25 \cos^{2}{\left(\frac{2 x}{5} + \frac{3 \pi}{10} \right)}}$

The answer is:

$- \frac{12}{25 \cos^{2}{\left(\frac{2 x}{5} + \frac{3 \pi}{10} \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

             2/2*x - pi\
       12*cot |--------|
  12          \   5    /
- -- - -----------------
  25           25

$$- \frac{12 \cot^{2}{\left(\frac{2 x - \pi}{5} \right)}}{25} - \frac{12}{25}$$

Simplify

The second derivative [src]

   /       2/-pi + 2*x\\    /-pi + 2*x\
48*|1 + cot |---------||*cot|---------|
   \        \    5    //    \    5    /
---------------------------------------
                  125

$$\frac{48 \left(\cot^{2}{\left(\frac{2 x - \pi}{5} \right)} + 1\right) \cot{\left(\frac{2 x - \pi}{5} \right)}}{125}$$

Simplify

The third derivative [src]

    /       2/-pi + 2*x\\ /         2/-pi + 2*x\\
-96*|1 + cot |---------||*|1 + 3*cot |---------||
    \        \    5    // \          \    5    //
-------------------------------------------------
                       625

$$- \frac{96 \left(\cot^{2}{\left(\frac{2 x - \pi}{5} \right)} + 1\right) \left(3 \cot^{2}{\left(\frac{2 x - \pi}{5} \right)} + 1\right)}{625}$$

Simplify

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Derivative of 6cot((2x-pi)/5)/5

The graph:

Piecewise:

The solution

Method #1

Method #2

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How to use it?

Identical expressions

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Derivative of 6*cot((2*x-pi)/5)/5

The graph:

Piecewise:

The solution

Method #1

Method #2

Derivative of 6cot((2x-pi)/5)/5