Derivative of ln(1-sin(2x))

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log(1 - sin(2*x))

$$\log{\left(- \sin{\left(2 x \right)} + 1 \right)}$$

d                    
--(log(1 - sin(2*x)))
dx

$$\frac{d}{d x} \log{\left(- \sin{\left(2 x \right)} + 1 \right)}$$

Transform

Detail solution

Let $u = 1 - \sin{\left(2 x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - \sin{\left(2 x \right)}\right)$ :
1. Differentiate $1 - \sin{\left(2 x \right)}$ 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 $u = 2 x$ .
    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 $\frac{d}{d x} 2 x$ :
      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: $2$
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
    So, the result is: $- 2 \cos{\left(2 x \right)}$
  The result is: $- 2 \cos{\left(2 x \right)}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /            2                        \         
  |       2*cos (2*x)        3*sin(2*x) |         
8*|-1 + ---------------- + -------------|*cos(2*x)
  |                    2   -1 + sin(2*x)|         
  \     (-1 + sin(2*x))                 /         
--------------------------------------------------
                  -1 + sin(2*x)

$$\frac{8 \left(-1 + \frac{3 \sin{\left(2 x \right)}}{\sin{\left(2 x \right)} - 1} + \frac{2 \cos^{2}{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} - 1\right)^{2}}\right) \cos{\left(2 x \right)}}{\sin{\left(2 x \right)} - 1}$$

Simplify

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Derivative of ln(1-sin(2x))

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