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Identical expressions
ln(one -x*cos(x))
ln(1 minus x multiply by co sinus of e of (x))
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ln1-xcosx
Similar expressions
ln(1+x*cos(x))
ln(1-x*cosx)

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

Derivative of ln(1-x*cos(x))

The solution

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log(1 - x*cos(x))

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

log(1 - x*cos(x))

Detail solution

Let $u = - x \cos{\left(x \right)} + 1$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x \cos{\left(x \right)} + 1\right)$ :
1. Differentiate $- x \cos{\left(x \right)} + 1$ 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. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      $g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      The result is: $- x \sin{\left(x \right)} + \cos{\left(x \right)}$
    So, the result is: $x \sin{\left(x \right)} - \cos{\left(x \right)}$
  The result is: $x \sin{\left(x \right)} - \cos{\left(x \right)}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(1-x*cos(x))

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The solution