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Derivative of:
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Identical expressions
y=(ctgx)/(x-x^ three)
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y=(ctgx)/(x-x3)
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Similar expressions
y=(ctgx)/(x+x^3)

The derivative of the function/ y=(ctgx)/(x-x^3)

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

The solution

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cot(x)
------
     3
x - x

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

cot(x)/(x - x^3)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \cot{\left(x \right)}$ and $g{\left(x \right)} = - x^{3} + x$ .

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

$\frac{- \left(1 - 3 x^{2}\right) \cot{\left(x \right)} - \frac{\left(- x^{3} + x\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}}{\left(- x^{3} + x\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2      /        2\       
-1 - cot (x)   \-1 + 3*x /*cot(x)
------------ + ------------------
        3                  2     
   x - x           /     3\      
                   \x - x /

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

Simplify

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 /

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

Simplify

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 /

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

Simplify

The graph

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

The graph:

Piecewise:

The solution

Method #1

Method #2