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Identical expressions
cot(x)/(x^ four - sixteen)
cotangent of (x) divide by (x to the power of 4 minus 16)
cotangent of (x) divide by (x to the power of four minus sixteen)
cot(x)/(x4-16)
cotx/x4-16
cot(x)/(x⁴-16)
cotx/x^4-16
cot(x) divide by (x^4-16)
Similar expressions
cot(x)/(x^4+16)

The derivative of the function/ cot(x)/(x^4-16)

Derivative of cot(x)/(x^4-16)

The solution

You have entered [src]

 cot(x)
-------
 4     
x  - 16

$$\frac{\cot{\left(x \right)}}{x^{4} - 16}$$

cot(x)/(x^4 - 16)

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 $g{\left(x \right)} = x^{4} - 16$ .

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 $x^{4} - 16$ term by term:
  1. The derivative of the constant $-16$ is zero.
  2. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
  The result is: $4 x^{3}$
Now plug in to the quotient rule:

$\frac{- 4 x^{3} \cot{\left(x \right)} - \frac{\left(x^{4} - 16\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}}{\left(x^{4} - 16\right)^{2}}$
Now simplify:

$\frac{2 \left(- x^{4} - 2 x^{3} \sin{\left(2 x \right)} + 16\right)}{\left(1 - \cos{\left(2 x \right)}\right) \left(x^{4} - 16\right)^{2}}$

The answer is:

$\frac{2 \left(- x^{4} - 2 x^{3} \sin{\left(2 x \right)} + 16\right)}{\left(1 - \cos{\left(2 x \right)}\right) \left(x^{4} - 16\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2         3       
-1 - cot (x)   4*x *cot(x)
------------ - -----------
   4                     2
  x  - 16       / 4     \ 
                \x  - 16/

$$- \frac{4 x^{3} \cot{\left(x \right)}}{\left(x^{4} - 16\right)^{2}} + \frac{- \cot^{2}{\left(x \right)} - 1}{x^{4} - 16}$$

Simplify

The second derivative [src]

  /                                                 /          4  \       \
  |                                               2 |       8*x   |       |
  |                                            2*x *|-3 + --------|*cot(x)|
  |                          3 /       2   \        |            4|       |
  |/       2   \          4*x *\1 + cot (x)/        \     -16 + x /       |
2*|\1 + cot (x)/*cot(x) + ------------------ + ---------------------------|
  |                                   4                         4         |
  \                            -16 + x                   -16 + x          /
---------------------------------------------------------------------------
                                         4                                 
                                  -16 + x

$$\frac{2 \left(\frac{4 x^{3} \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{4} - 16} + \frac{2 x^{2} \left(\frac{8 x^{4}}{x^{4} - 16} - 3\right) \cot{\left(x \right)}}{x^{4} - 16} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)}{x^{4} - 16}$$

Simplify

The third derivative [src]

   /                                                                          /         4            8   \                                    \
   |                                                   /          4  \        |     12*x         16*x    |                                    |
   |                                   2 /       2   \ |       8*x   |   12*x*|1 - -------- + -----------|*cot(x)                             |
   |                                6*x *\1 + cot (x)/*|-3 + --------|        |           4             2|                                    |
   |                                                   |            4|        |    -16 + x    /       4\ |              3 /       2   \       |
   |/       2   \ /         2   \                      \     -16 + x /        \               \-16 + x / /          12*x *\1 + cot (x)/*cot(x)|
-2*|\1 + cot (x)/*\1 + 3*cot (x)/ + ---------------------------------- + ---------------------------------------- + --------------------------|
   |                                                    4                                       4                                   4         |
   \                                             -16 + x                                 -16 + x                             -16 + x          /
-----------------------------------------------------------------------------------------------------------------------------------------------
                                                                           4                                                                   
                                                                    -16 + x

$$- \frac{2 \left(\frac{12 x^{3} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x^{4} - 16} + \frac{6 x^{2} \left(\frac{8 x^{4}}{x^{4} - 16} - 3\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{4} - 16} + \frac{12 x \left(\frac{16 x^{8}}{\left(x^{4} - 16\right)^{2}} - \frac{12 x^{4}}{x^{4} - 16} + 1\right) \cot{\left(x \right)}}{x^{4} - 16} + \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right)\right)}{x^{4} - 16}$$

Simplify

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Derivative of cot(x)/(x^4-16)

The graph:

Piecewise:

The solution

Method #1

Method #2