Derivative of (tan2x)/(1-cot2x)

The solution

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  tan(2*x)  
------------
1 - cot(2*x)

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

d /  tan(2*x)  \
--|------------|
dx\1 - cot(2*x)/

$$\frac{d}{d x} \frac{\tan{\left(2 x \right)}}{1 - \cot{\left(2 x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \tan{\left(2 x \right)}$ and $g{\left(x \right)} = 1 - \cot{\left(2 x \right)}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                                                                               /         2                \         \
  |                                                               /       2     \ |  1 + cot (2*x)           |         |
  |                             /       2     \ /       2     \   \1 + cot (2*x)/*|- ------------- + cot(2*x)|*tan(2*x)|
  |  /       2     \            \1 + cot (2*x)/*\1 + tan (2*x)/                   \  -1 + cot(2*x)           /         |
8*|- \1 + tan (2*x)/*tan(2*x) - ------------------------------- + -----------------------------------------------------|
  \                                      -1 + cot(2*x)                                -1 + cot(2*x)                    /
------------------------------------------------------------------------------------------------------------------------
                                                     -1 + cot(2*x)

$$\frac{8 \left(- \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} - \frac{\left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right)}{\cot{\left(2 x \right)} - 1} + \frac{\left(\cot{\left(2 x \right)} - \frac{\cot^{2}{\left(2 x \right)} + 1}{\cot{\left(2 x \right)} - 1}\right) \left(\cot^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)}}{\cot{\left(2 x \right)} - 1}\right)}{\cot{\left(2 x \right)} - 1}$$

Simplify

The third derivative [src]

   /                                                      /                                   2                             \                                                                                                                       \
   |                                                      |                    /       2     \      /       2     \         |                                                                                                                       |
   |                                      /       2     \ |         2        3*\1 + cot (2*x)/    6*\1 + cot (2*x)/*cot(2*x)|                                                                                           /         2                \|
   |                                      \1 + cot (2*x)/*|1 + 3*cot (2*x) + ------------------ - --------------------------|*tan(2*x)                                                  /       2     \ /       2     \ |  1 + cot (2*x)           ||
   |                                                      |                                  2          -1 + cot(2*x)       |              /       2     \ /       2     \            3*\1 + cot (2*x)/*\1 + tan (2*x)/*|- ------------- + cot(2*x)||
   |  /       2     \ /         2     \                   \                   (-1 + cot(2*x))                               /            3*\1 + cot (2*x)/*\1 + tan (2*x)/*tan(2*x)                                     \  -1 + cot(2*x)           /|
16*|- \1 + tan (2*x)/*\1 + 3*tan (2*x)/ - -------------------------------------------------------------------------------------------- - ------------------------------------------ + --------------------------------------------------------------|
   \                                                                             -1 + cot(2*x)                                                         -1 + cot(2*x)                                          -1 + cot(2*x)                         /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                    -1 + cot(2*x)

$$\frac{16 \left(- \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(3 \tan^{2}{\left(2 x \right)} + 1\right) + \frac{3 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot{\left(2 x \right)} - \frac{\cot^{2}{\left(2 x \right)} + 1}{\cot{\left(2 x \right)} - 1}\right) \left(\cot^{2}{\left(2 x \right)} + 1\right)}{\cot{\left(2 x \right)} - 1} - \frac{3 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)}}{\cot{\left(2 x \right)} - 1} - \frac{\left(\cot^{2}{\left(2 x \right)} + 1\right) \left(3 \cot^{2}{\left(2 x \right)} + 1 - \frac{6 \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)}}{\cot{\left(2 x \right)} - 1} + \frac{3 \left(\cot^{2}{\left(2 x \right)} + 1\right)^{2}}{\left(\cot{\left(2 x \right)} - 1\right)^{2}}\right) \tan{\left(2 x \right)}}{\cot{\left(2 x \right)} - 1}\right)}{\cot{\left(2 x \right)} - 1}$$

Simplify

The graph

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Identical expressions

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Derivative of (tan2x)/(1-cot2x)

The graph:

Piecewise:

The solution

Method #1

Method #2