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Derivative of ctgx/(x^2+1)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
cot(x)
------
 2    
x  + 1
$$\frac{\cot{\left(x \right)}}{x^{2} + 1}$$
cot(x)/(x^2 + 1)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. There are multiple ways to do this derivative.

      Method #1

      1. Rewrite the function to be differentiated:

      2. Let .

      3. Apply the power rule: goes to

      4. Then, apply the chain rule. Multiply by :

        1. Rewrite the function to be differentiated:

        2. Apply the quotient rule, which is:

          and .

          To find :

          1. The derivative of sine is cosine:

          To find :

          1. The derivative of cosine is negative sine:

          Now plug in to the quotient rule:

        The result of the chain rule is:

      Method #2

      1. Rewrite the function to be differentiated:

      2. Apply the quotient rule, which is:

        and .

        To find :

        1. The derivative of cosine is negative sine:

        To find :

        1. The derivative of sine is cosine:

        Now plug in to the quotient rule:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. Apply the power rule: goes to

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
        2                
-1 - cot (x)   2*x*cot(x)
------------ - ----------
    2                  2 
   x  + 1      / 2    \  
               \x  + 1/  
$$- \frac{2 x \cot{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{- \cot^{2}{\left(x \right)} - 1}{x^{2} + 1}$$
The second derivative [src]
  /                       /         2 \                           \
  |                       |      4*x  |                           |
  |                       |-1 + ------|*cot(x)                    |
  |                       |          2|              /       2   \|
  |/       2   \          \     1 + x /          2*x*\1 + cot (x)/|
2*|\1 + cot (x)/*cot(x) + -------------------- + -----------------|
  |                                   2                     2     |
  \                              1 + x                 1 + x      /
-------------------------------------------------------------------
                                    2                              
                               1 + x                               
$$\frac{2 \left(\frac{2 x \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2} + 1} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \frac{\left(\frac{4 x^{2}}{x^{2} + 1} - 1\right) \cot{\left(x \right)}}{x^{2} + 1}\right)}{x^{2} + 1}$$
The third derivative [src]
   /                                                /         2 \                                   /         2 \       \
   |                                  /       2   \ |      4*x  |                                   |      2*x  |       |
   |                                3*\1 + cot (x)/*|-1 + ------|                              12*x*|-1 + ------|*cot(x)|
   |                                                |          2|       /       2   \               |          2|       |
   |/       2   \ /         2   \                   \     1 + x /   6*x*\1 + cot (x)/*cot(x)        \     1 + x /       |
-2*|\1 + cot (x)/*\1 + 3*cot (x)/ + ----------------------------- + ------------------------ + -------------------------|
   |                                                 2                            2                            2        |
   |                                            1 + x                        1 + x                     /     2\         |
   \                                                                                                   \1 + x /         /
-------------------------------------------------------------------------------------------------------------------------
                                                               2                                                         
                                                          1 + x                                                          
$$- \frac{2 \left(\frac{6 x \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x^{2} + 1} + \frac{12 x \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right) \cot{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right) + \frac{3 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2} + 1}\right)}{x^{2} + 1}$$