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1/cos2x-1
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Similar expressions
1/(cos(2*x)+1)
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The derivative of the function/ 1/(cos(2*x)-1)

Derivative of 1/(cos(2*x)-1)

The solution

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

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

1/(cos(2*x) - 1)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                            2        \         
  |       6*cos(2*x)      6*sin (2*x)   |         
8*|-1 + ------------- + ----------------|*sin(2*x)
  |     -1 + cos(2*x)                  2|         
  \                     (-1 + cos(2*x)) /         
--------------------------------------------------
                                2                 
                 (-1 + cos(2*x))

$$\frac{8 \left(-1 + \frac{6 \cos{\left(2 x \right)}}{\cos{\left(2 x \right)} - 1} + \frac{6 \sin^{2}{\left(2 x \right)}}{\left(\cos{\left(2 x \right)} - 1\right)^{2}}\right) \sin{\left(2 x \right)}}{\left(\cos{\left(2 x \right)} - 1\right)^{2}}$$

Simplify

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Derivative of 1/(cos(2*x)-1)

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The solution