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Derivative of ctg(x)/x^4

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
cot(x)
------
   4  
  x   
$$\frac{\cot{\left(x \right)}}{x^{4}}$$
cot(x)/x^4
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. There are multiple ways to do this derivative.

      Method #1

      1. Rewrite the function to be differentiated:

      2. Let .

      3. Apply the power rule: goes to

      4. Then, apply the chain rule. Multiply by :

        1. Rewrite the function to be differentiated:

        2. Apply the quotient rule, which is:

          and .

          To find :

          1. The derivative of sine is cosine:

          To find :

          1. The derivative of cosine is negative sine:

          Now plug in to the quotient rule:

        The result of the chain rule is:

      Method #2

      1. Rewrite the function to be differentiated:

      2. Apply the quotient rule, which is:

        and .

        To find :

        1. The derivative of cosine is negative sine:

        To find :

        1. The derivative of sine is cosine:

        Now plug in to the quotient rule:

    To find :

    1. Apply the power rule: goes to

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
        2              
-1 - cot (x)   4*cot(x)
------------ - --------
      4            5   
     x            x    
$$\frac{- \cot^{2}{\left(x \right)} - 1}{x^{4}} - \frac{4 \cot{\left(x \right)}}{x^{5}}$$
The second derivative [src]
  /                         /       2   \            \
  |/       2   \          4*\1 + cot (x)/   10*cot(x)|
2*|\1 + cot (x)/*cot(x) + --------------- + ---------|
  |                              x               2   |
  \                                             x    /
------------------------------------------------------
                           4                          
                          x                           
$$\frac{2 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{4 \left(\cot^{2}{\left(x \right)} + 1\right)}{x} + \frac{10 \cot{\left(x \right)}}{x^{2}}\right)}{x^{4}}$$
The third derivative [src]
   /                                   /       2   \                  /       2   \       \
   |/       2   \ /         2   \   30*\1 + cot (x)/   60*cot(x)   12*\1 + cot (x)/*cot(x)|
-2*|\1 + cot (x)/*\1 + 3*cot (x)/ + ---------------- + --------- + -----------------------|
   |                                        2               3                 x           |
   \                                       x               x                              /
-------------------------------------------------------------------------------------------
                                              4                                            
                                             x                                             
$$- \frac{2 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right) + \frac{12 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x} + \frac{30 \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2}} + \frac{60 \cot{\left(x \right)}}{x^{3}}\right)}{x^{4}}$$