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Derivative of:
Derivative of arctg(1/x)
Derivative of e^cos(x)
Derivative of ln(x+3)^3
Derivative of cot(1/x)
Identical expressions
4ctg((x/ two)+(pi/ six))
4ctg((x divide by 2) plus ( Pi divide by 6))
4ctg((x divide by two) plus ( Pi divide by six))
4ctgx/2+pi/6
4ctg((x divide by 2)+(pi divide by 6))
Similar expressions
4ctg((x/2)-(pi/6))

The derivative of the function/ 4ctg((x/2)+(pi/6))

Derivative of 4ctg((x/2)+(pi/6))

The solution

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     /x   pi\
4*cot|- + --|
     \2   6 /

$$4 \cot{\left(\frac{x}{2} + \frac{\pi}{6} \right)}$$

4*cot(x/2 + pi/6)

Detail solution

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

$- \frac{4}{\left(\cos{\left(x + \frac{\pi}{3} \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{6} \right)}}$

The answer is:

$- \frac{4}{\left(\cos{\left(x + \frac{\pi}{3} \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{6} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of 4ctg((x/2)+(pi/6))

The graph:

Piecewise:

The solution

Method #1

Method #2