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ctgx/(x^2-1)

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

Derivative of ctgx/(x^2+1)

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

Apply the quotient rule, which is:

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

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

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:
    
    $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
  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:
    
    $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

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

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

Simplify

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

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

Simplify

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Derivative of ctgx/(x^2+1)

The graph:

Piecewise:

The solution

Method #1

Method #2