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Derivative of:
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Identical expressions
one /(one -cos(3x))
1 divide by (1 minus co sinus of e of (3x))
one divide by (one minus co sinus of e of (3x))
1/1-cos3x
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Similar expressions
1/(1+cos(3x))

The derivative of the function/ 1/(1-cos(3x))

Derivative of 1/(1-cos(3x))

The solution

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     1      
------------
1 - cos(3*x)

$$\frac{1}{1 - \cos{\left(3 x \right)}}$$

1/(1 - cos(3*x))

Detail solution

Let $u = 1 - \cos{\left(3 x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - \cos{\left(3 x \right)}\right)$ :
1. Differentiate $1 - \cos{\left(3 x \right)}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 3 x$ .
    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 $\frac{d}{d x} 3 x$ :
      1. 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: $3$
      The result of the chain rule is:
      
      $- 3 \sin{\left(3 x \right)}$
    So, the result is: $3 \sin{\left(3 x \right)}$
  The result is: $3 \sin{\left(3 x \right)}$
The result of the chain rule is:

$- \frac{3 \sin{\left(3 x \right)}}{\left(1 - \cos{\left(3 x \right)}\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  -3*sin(3*x)  
---------------
              2
(1 - cos(3*x))

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

Simplify

The second derivative [src]

   /      2                 \
   | 2*sin (3*x)            |
-9*|------------- + cos(3*x)|
   \-1 + cos(3*x)           /
-----------------------------
                      2      
       (-1 + cos(3*x))

$$- \frac{9 \left(\cos{\left(3 x \right)} + \frac{2 \sin^{2}{\left(3 x \right)}}{\cos{\left(3 x \right)} - 1}\right)}{\left(\cos{\left(3 x \right)} - 1\right)^{2}}$$

Simplify

The third derivative [src]

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

$$\frac{27 \left(1 - \frac{6 \cos{\left(3 x \right)}}{\cos{\left(3 x \right)} - 1} - \frac{6 \sin^{2}{\left(3 x \right)}}{\left(\cos{\left(3 x \right)} - 1\right)^{2}}\right) \sin{\left(3 x \right)}}{\left(\cos{\left(3 x \right)} - 1\right)^{2}}$$

Simplify

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Derivative of 1/(1-cos(3x))

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The solution