Derivative of (ctg(x)+x)/(1-xctg(x))

The solution

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 cot(x) + x 
------------
1 - x*cot(x)

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

(cot(x) + x)/(1 - x*cot(x))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x + \cot{\left(x \right)}$ and $g{\left(x \right)} = - x \cot{\left(x \right)} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x + \cot{\left(x \right)}$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. 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 $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
      
      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 $\frac{d}{d x} f{\left(x \right)}$ :
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      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:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
  The result is: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + 1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $- x \cot{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      $g{\left(x \right)} = \cot{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. $\frac{d}{d x} \cot{\left(x \right)} = - \frac{1}{\sin^{2}{\left(x \right)}}$
      The result is: $- \frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \cot{\left(x \right)}$
    So, the result is: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \cot{\left(x \right)}$
  The result is: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \cot{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (ctg(x)+x)/(1-xctg(x))

The graph:

Piecewise:

The solution

Method #1

Method #2