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Identical expressions
cot(two *x)/cos(six *x)
cotangent of (2 multiply by x) divide by co sinus of e of (6 multiply by x)
cotangent of (two multiply by x) divide by co sinus of e of (six multiply by x)
cot(2x)/cos(6x)
cot2x/cos6x
cot(2*x) divide by cos(6*x)

The derivative of the function/ cot(2*x)/cos(6*x)

Derivative of cot(2x)/cos(6x)

The solution

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cot(2*x)
--------
cos(6*x)

$$\frac{\cot{\left(2 x \right)}}{\cos{\left(6 x \right)}}$$

cot(2*x)/cos(6*x)

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 $g{\left(x \right)} = \cos{\left(6 x \right)}$ .

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. Let $u = 6 x$ .
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 $\frac{d}{d x} 6 x$ :
  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: $6$
  The result of the chain rule is:
  
  $- 6 \sin{\left(6 x \right)}$
Now plug in to the quotient rule:

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

$\frac{2 \left(\frac{3 \cos{\left(2 x \right)}}{4} - \cos{\left(6 x \right)} - \frac{3 \cos{\left(10 x \right)}}{4}\right)}{\sin^{2}{\left(2 x \right)} \cos^{2}{\left(6 x \right)}}$

The answer is:

$\frac{2 \left(\frac{3 \cos{\left(2 x \right)}}{4} - \cos{\left(6 x \right)} - \frac{3 \cos{\left(10 x \right)}}{4}\right)}{\sin^{2}{\left(2 x \right)} \cos^{2}{\left(6 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          2                           
-2 - 2*cot (2*x)   6*cot(2*x)*sin(6*x)
---------------- + -------------------
    cos(6*x)               2          
                        cos (6*x)

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

Simplify

The second derivative [src]

  /                               /         2     \              /       2     \         \
  |  /       2     \              |    2*sin (6*x)|            6*\1 + cot (2*x)/*sin(6*x)|
4*|2*\1 + cot (2*x)/*cot(2*x) + 9*|1 + -----------|*cot(2*x) - --------------------------|
  |                               |        2      |                     cos(6*x)         |
  \                               \     cos (6*x) /                                      /
------------------------------------------------------------------------------------------
                                         cos(6*x)

$$\frac{4 \left(9 \left(\frac{2 \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 1\right) \cot{\left(2 x \right)} - \frac{6 \left(\cot^{2}{\left(2 x \right)} + 1\right) \sin{\left(6 x \right)}}{\cos{\left(6 x \right)}} + 2 \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)}\right)}{\cos{\left(6 x \right)}}$$

Simplify

The third derivative [src]

  /                                                                                                                         /         2     \                  \
  |                                                                                                                         |    6*sin (6*x)|                  |
  |                                                                                                                      27*|5 + -----------|*cot(2*x)*sin(6*x)|
  |                     /         2     \                                            /       2     \                        |        2      |                  |
  |     /       2     \ |    2*sin (6*x)|     /       2     \ /         2     \   18*\1 + cot (2*x)/*cot(2*x)*sin(6*x)      \     cos (6*x) /                  |
8*|- 27*\1 + cot (2*x)/*|1 + -----------| - 2*\1 + cot (2*x)/*\1 + 3*cot (2*x)/ + ------------------------------------ + --------------------------------------|
  |                     |        2      |                                                       cos(6*x)                                cos(6*x)               |
  \                     \     cos (6*x) /                                                                                                                      /
----------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                            cos(6*x)

$$\frac{8 \left(- 27 \left(\frac{2 \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right) + \frac{27 \left(\frac{6 \sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 5\right) \sin{\left(6 x \right)} \cot{\left(2 x \right)}}{\cos{\left(6 x \right)}} - 2 \left(\cot^{2}{\left(2 x \right)} + 1\right) \left(3 \cot^{2}{\left(2 x \right)} + 1\right) + \frac{18 \left(\cot^{2}{\left(2 x \right)} + 1\right) \sin{\left(6 x \right)} \cot{\left(2 x \right)}}{\cos{\left(6 x \right)}}\right)}{\cos{\left(6 x \right)}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of cot(2x)/cos(6x)

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Derivative of cot(2*x)/cos(6*x)

The graph:

Piecewise:

The solution

Method #1

Method #2

Derivative of cot(2x)/cos(6x)