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y=(ctgx)/(x-x^3)

Derivative of y=(ctgx)/(x-x^3)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
cot(x)
------
     3
x - x 
$$\frac{\cot{\left(x \right)}}{- x^{3} + x}$$
cot(x)/(x - x^3)
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. Differentiate term by term:

      1. Apply the power rule: goes to

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
        2      /        2\       
-1 - cot (x)   \-1 + 3*x /*cot(x)
------------ + ------------------
        3                  2     
   x - x           /     3\      
                   \x - x /      
$$\frac{\left(3 x^{2} - 1\right) \cot{\left(x \right)}}{\left(- x^{3} + x\right)^{2}} + \frac{- \cot^{2}{\left(x \right)} - 1}{- x^{3} + x}$$
The second derivative [src]
  /                         /               2\                                   \
  |                         |    /        2\ |                                   |
  |                         |    \-1 + 3*x / |                                   |
  |                         |3 - ------------|*cot(x)                            |
  |                         |     2 /      2\|          /       2   \ /        2\|
  |  /       2   \          \    x *\-1 + x //          \1 + cot (x)/*\-1 + 3*x /|
2*|- \1 + cot (x)/*cot(x) + ------------------------- - -------------------------|
  |                                        2                     /      2\       |
  \                                  -1 + x                    x*\-1 + x /       /
----------------------------------------------------------------------------------
                                     /      2\                                    
                                   x*\-1 + x /                                    
$$\frac{2 \left(\frac{\left(3 - \frac{\left(3 x^{2} - 1\right)^{2}}{x^{2} \left(x^{2} - 1\right)}\right) \cot{\left(x \right)}}{x^{2} - 1} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{\left(3 x^{2} - 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{x \left(x^{2} - 1\right)}\right)}{x \left(x^{2} - 1\right)}$$
The third derivative [src]
  /                                                                       /                                3\                                            \
  |                                                /               2\     |      /        2\    /        2\ |                                            |
  |                                                |    /        2\ |     |    6*\-1 + 3*x /    \-1 + 3*x / |                                            |
  |                                  /       2   \ |    \-1 + 3*x / |   3*|1 - ------------- + -------------|*cot(x)                                     |
  |                                3*\1 + cot (x)/*|3 - ------------|     |             2                  2|                                            |
  |                                                |     2 /      2\|     |       -1 + x        2 /      2\ |            /       2   \ /        2\       |
  |/       2   \ /         2   \                   \    x *\-1 + x //     \                    x *\-1 + x / /          3*\1 + cot (x)/*\-1 + 3*x /*cot(x)|
2*|\1 + cot (x)/*\1 + 3*cot (x)/ - ---------------------------------- + -------------------------------------------- + ----------------------------------|
  |                                                   2                                   /      2\                                 /      2\            |
  \                                             -1 + x                                  x*\-1 + x /                               x*\-1 + x /            /
----------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                         /      2\                                                                        
                                                                       x*\-1 + x /                                                                        
$$\frac{2 \left(- \frac{3 \left(3 - \frac{\left(3 x^{2} - 1\right)^{2}}{x^{2} \left(x^{2} - 1\right)}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2} - 1} + \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right) + \frac{3 \left(3 x^{2} - 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x \left(x^{2} - 1\right)} + \frac{3 \left(1 - \frac{6 \left(3 x^{2} - 1\right)}{x^{2} - 1} + \frac{\left(3 x^{2} - 1\right)^{3}}{x^{2} \left(x^{2} - 1\right)^{2}}\right) \cot{\left(x \right)}}{x \left(x^{2} - 1\right)}\right)}{x \left(x^{2} - 1\right)}$$
The graph
Derivative of y=(ctgx)/(x-x^3)