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ctg(2x)/(1-sin^2(x))

The derivative of the function/ ctg(2x)/(1+sin^2(x))

Derivative of ctg(2x)/(1+sin^2(x))

The solution

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  cot(2*x) 
-----------
       2   
1 + sin (x)

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cot{\left(2 x \right)}$ and $g{\left(x \right)} = \sin^{2}{\left(x \right)} + 1$ .

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

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

$\frac{2 \cdot \left(3 \cos{\left(2 x \right)} + \cos{\left(6 x \right)} - 12\right)}{\left(1 - \cos{\left(4 x \right)}\right) \left(3 - \cos{\left(2 x \right)}\right)^{2}}$

The answer is:

$\frac{2 \cdot \left(3 \cos{\left(2 x \right)} + \cos{\left(6 x \right)} - 12\right)}{\left(1 - \cos{\left(4 x \right)}\right) \left(3 - \cos{\left(2 x \right)}\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                             /                         2       2   \                                           \
  |                             |   2         2      4*cos (x)*sin (x)|                                           |
  |                             |sin (x) - cos (x) + -----------------|*cot(2*x)                                  |
  |                             |                              2      |              /       2     \              |
  |  /       2     \            \                       1 + sin (x)   /            4*\1 + cot (2*x)/*cos(x)*sin(x)|
2*|4*\1 + cot (2*x)/*cot(2*x) + ------------------------------------------------ + -------------------------------|
  |                                                      2                                          2             |
  \                                               1 + sin (x)                                1 + sin (x)          /
-------------------------------------------------------------------------------------------------------------------
                                                           2                                                       
                                                    1 + sin (x)

$$\frac{2 \cdot \left(4 \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)} + \frac{4 \left(\cot^{2}{\left(2 x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{\left(\frac{4 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cot{\left(2 x \right)}}{\sin^{2}{\left(x \right)} + 1}\right)}{\sin^{2}{\left(x \right)} + 1}$$

Simplify

The third derivative [src]

  /                                                                                                                                                  /          2             2            2       2   \                       \
  |                                                          /                         2       2   \                                                 |     3*sin (x)     3*cos (x)    6*cos (x)*sin (x)|                       |
  |                                          /       2     \ |   2         2      4*cos (x)*sin (x)|                                               2*|1 - ----------- + ----------- - -----------------|*cos(x)*cot(2*x)*sin(x)|
  |                                        3*\1 + cot (2*x)/*|sin (x) - cos (x) + -----------------|                                                 |           2             2                     2 |                       |
  |                                                          |                              2      |      /       2     \                            |    1 + sin (x)   1 + sin (x)     /       2   \  |                       |
  |    /       2     \ /         2     \                     \                       1 + sin (x)   /   12*\1 + cot (2*x)/*cos(x)*cot(2*x)*sin(x)     \                                  \1 + sin (x)/  /                       |
4*|- 4*\1 + cot (2*x)/*\1 + 3*cot (2*x)/ - --------------------------------------------------------- - ----------------------------------------- + ----------------------------------------------------------------------------|
  |                                                                      2                                                   2                                                            2                                    |
  \                                                               1 + sin (x)                                         1 + sin (x)                                                  1 + sin (x)                                 /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                 2                                                                                                              
                                                                                                          1 + sin (x)

$$\frac{4 \left(- \frac{12 \left(\cot^{2}{\left(2 x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} \cot{\left(2 x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{2 \cdot \left(- \frac{6 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\sin^{2}{\left(x \right)} + 1\right)^{2}} - \frac{3 \sin^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} \cot{\left(2 x \right)}}{\sin^{2}{\left(x \right)} + 1} - 4 \left(\cot^{2}{\left(2 x \right)} + 1\right) \left(3 \cot^{2}{\left(2 x \right)} + 1\right) - \frac{3 \left(\cot^{2}{\left(2 x \right)} + 1\right) \left(\frac{4 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)} + 1}\right)}{\sin^{2}{\left(x \right)} + 1}$$

Simplify

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Derivative of ctg(2x)/(1+sin^2(x))

The graph:

Piecewise:

The solution

Method #1

Method #2