Derivative of y=ctgx+x/(1-xctgx)

The solution

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

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

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

Detail solution

Differentiate $\frac{x}{- x \cot{\left(x \right)} + 1} + \cot{\left(x \right)}$ term by term:
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:
    
    $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x$ and $g{\left(x \right)} = - x \cot{\left(x \right)} + 1$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  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 $\frac{d}{d x} f{\left(x \right)}$ :
        
        Apply the power rule: $x$ goes to $1$
        
        $g{\left(x \right)} = \cot{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
        
        $\frac{d}{d x} \cot{\left(x \right)} = - \frac{1}{\sin^{2}{\left(x \right)}}$
        
        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:
  
  $\frac{- x \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \cot{\left(x \right)}\right) - x \cot{\left(x \right)} + 1}{\left(- x \cot{\left(x \right)} + 1\right)^{2}}$
The result is: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{- x \left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \cot{\left(x \right)}\right) - x \cot{\left(x \right)} + 1}{\left(- x \cot{\left(x \right)} + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /                                                                        2                                                                             3                                                                                                                                       \
   |             2                               /            /       2   \\      /       2        /       2   \       \       /            /       2   \\      /       2   \ /              /       2   \          2   \       /            /       2   \\ /       2        /       2   \       \|
   |/       2   \         2    /       2   \   3*\-cot(x) + x*\1 + cot (x)//    3*\1 + cot (x) - x*\1 + cot (x)/*cot(x)/   3*x*\-cot(x) + x*\1 + cot (x)//    x*\1 + cot (x)/*\-3*cot(x) + x*\1 + cot (x)/ + 2*x*cot (x)/   6*x*\-cot(x) + x*\1 + cot (x)//*\1 + cot (x) - x*\1 + cot (x)/*cot(x)/|
-2*|\1 + cot (x)/  + 2*cot (x)*\1 + cot (x)/ + ------------------------------ + ---------------------------------------- + -------------------------------- + ----------------------------------------------------------- + ----------------------------------------------------------------------|
   |                                                                 3                                     2                                      4                                                2                                                                  3                           |
   \                                                  (-1 + x*cot(x))                       (-1 + x*cot(x))                        (-1 + x*cot(x))                                  (-1 + x*cot(x))                                                    (-1 + x*cot(x))                            /

$$- 2 \left(\frac{3 x \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right)^{3}}{\left(x \cot{\left(x \right)} - 1\right)^{4}} + \frac{6 x \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right) \left(- x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right)}{\left(x \cot{\left(x \right)} - 1\right)^{3}} + \frac{x \left(\cot^{2}{\left(x \right)} + 1\right) \left(x \left(\cot^{2}{\left(x \right)} + 1\right) + 2 x \cot^{2}{\left(x \right)} - 3 \cot{\left(x \right)}\right)}{\left(x \cot{\left(x \right)} - 1\right)^{2}} + \frac{3 \left(x \left(\cot^{2}{\left(x \right)} + 1\right) - \cot{\left(x \right)}\right)^{2}}{\left(x \cot{\left(x \right)} - 1\right)^{3}} + \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + \frac{3 \left(- x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right)}{\left(x \cot{\left(x \right)} - 1\right)^{2}}\right)$$

Simplify

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Derivative of y=ctgx+x/(1-xctgx)

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The solution

Method #1

Method #2