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Derivative of:
Derivative of x/log(x)
Derivative of x*e^x^2
Derivative of (x+1)/(x-1)
Derivative of (x+4)^2*(x+1)+9
Identical expressions
ctg^ three (x/ four)
ctg cubed (x divide by 4)
ctg to the power of three (x divide by four)
ctg3(x/4)
ctg3x/4
ctg³(x/4)
ctg to the power of 3(x/4)
ctg^3x/4
ctg^3(x divide by 4)

The derivative of the function/ ctg^3(x/4)

Derivative of ctg^3(x/4)

The solution

You have entered [src]

   3/x\
cot |-|
    \4/

$$\cot^{3}{\left(\frac{x}{4} \right)}$$

d /   3/x\\
--|cot |-||
dx\    \4//

$$\frac{d}{d x} \cot^{3}{\left(\frac{x}{4} \right)}$$

Transform

Detail solution

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

$- \frac{3 \left(\frac{\sin^{2}{\left(\frac{x}{4} \right)}}{4} + \frac{\cos^{2}{\left(\frac{x}{4} \right)}}{4}\right) \cot^{2}{\left(\frac{x}{4} \right)}}{\cos^{2}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{4} \right)}}$
Now simplify:

$- \frac{3 \cos^{2}{\left(\frac{x}{4} \right)}}{4 \sin^{4}{\left(\frac{x}{4} \right)}}$

The answer is:

$- \frac{3 \cos^{2}{\left(\frac{x}{4} \right)}}{4 \sin^{4}{\left(\frac{x}{4} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /           2/x\\
        |      3*cot |-||
   2/x\ |  3         \4/|
cot |-|*|- - - ---------|
    \4/ \  4       4    /

$$\left(- \frac{3 \cot^{2}{\left(\frac{x}{4} \right)}}{4} - \frac{3}{4}\right) \cot^{2}{\left(\frac{x}{4} \right)}$$

Simplify

The second derivative [src]

  /       2/x\\ /         2/x\\    /x\
3*|1 + cot |-||*|1 + 2*cot |-||*cot|-|
  \        \4// \          \4//    \4/
--------------------------------------
                  8

$$\frac{3 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right) \left(2 \cot^{2}{\left(\frac{x}{4} \right)} + 1\right) \cot{\left(\frac{x}{4} \right)}}{8}$$

Simplify

The third derivative [src]

                 /             2                                      \
   /       2/x\\ |/       2/x\\         4/x\        2/x\ /       2/x\\|
-3*|1 + cot |-||*||1 + cot |-||  + 2*cot |-| + 7*cot |-|*|1 + cot |-|||
   \        \4// \\        \4//          \4/         \4/ \        \4///
-----------------------------------------------------------------------
                                   32

$$- \frac{3 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right) \left(\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} + 7 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right) \cot^{2}{\left(\frac{x}{4} \right)} + 2 \cot^{4}{\left(\frac{x}{4} \right)}\right)}{32}$$

Simplify

The graph

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How to use it?

Identical expressions

Derivative of ctg^3(x/4)

The graph:

Piecewise:

The solution

Method #1

Method #2