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Derivative of:
Derivative of x^e
Derivative of (5*x-4)*(2*x^4-7*x+1)
Derivative of -3*x^3+2*x^2-x-5
Derivative of x/5+5/x
Identical expressions
cot(x)/((six *x))
cotangent of (x) divide by ((6 multiply by x))
cotangent of (x) divide by ((six multiply by x))
cot(x)/((6x))
cotx/6x
cot(x) divide by ((6*x))

The derivative of the function/ cot(x)/((6*x))

Derivative of cot(x)/((6*x))

The solution

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cot(x)
------
 6*x

$$\frac{\cot{\left(x \right)}}{6 x}$$

d /cot(x)\
--|------|
dx\ 6*x  /

$$\frac{d}{d x} \frac{\cot{\left(x \right)}}{6 x}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cot{\left(x \right)}$ and $g{\left(x \right)} = 6 x$ .

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. 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: $6$
Now plug in to the quotient rule:

$\frac{- \frac{6 x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - 6 \cot{\left(x \right)}}{36 x^{2}}$
Now simplify:

$\frac{2 x + \sin{\left(2 x \right)}}{6 x^{2} \left(\cos{\left(2 x \right)} - 1\right)}$

The answer is:

$\frac{2 x + \sin{\left(2 x \right)}}{6 x^{2} \left(\cos{\left(2 x \right)} - 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of cot(x)/((6*x))

The graph:

Piecewise:

The solution

Method #1

Method #2