Derivative of (lnx)/(4-3cosx)

The solution

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   log(x)   
------------
4 - 3*cos(x)

$$\frac{\log{\left(x \right)}}{4 - 3 \cos{\left(x \right)}}$$

d /   log(x)   \
--|------------|
dx\4 - 3*cos(x)/

$$\frac{d}{d x} \frac{\log{\left(x \right)}}{4 - 3 \cos{\left(x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = 4 - 3 \cos{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $4 - 3 \cos{\left(x \right)}$ term by term:
  1. The derivative of the constant $4$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $3 \sin{\left(x \right)}$
  The result is: $3 \sin{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{- 3 \log{\left(x \right)} \sin{\left(x \right)} + \frac{4 - 3 \cos{\left(x \right)}}{x}}{\left(4 - 3 \cos{\left(x \right)}\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       1           3*log(x)*sin(x)
---------------- - ---------------
x*(4 - 3*cos(x))                 2
                   (4 - 3*cos(x))

$$- \frac{3 \log{\left(x \right)} \sin{\left(x \right)}}{\left(4 - 3 \cos{\left(x \right)}\right)^{2}} + \frac{1}{x \left(4 - 3 \cos{\left(x \right)}\right)}$$

Simplify

The second derivative [src]

                           /       2              \       
                           |  6*sin (x)           |       
                         3*|------------- + cos(x)|*log(x)
1         6*sin(x)         \-4 + 3*cos(x)         /       
-- - ----------------- - ---------------------------------
 2   x*(-4 + 3*cos(x))             -4 + 3*cos(x)          
x                                                         
----------------------------------------------------------
                      -4 + 3*cos(x)

$$\frac{- \frac{3 \left(\cos{\left(x \right)} + \frac{6 \sin^{2}{\left(x \right)}}{3 \cos{\left(x \right)} - 4}\right) \log{\left(x \right)}}{3 \cos{\left(x \right)} - 4} - \frac{6 \sin{\left(x \right)}}{x \left(3 \cos{\left(x \right)} - 4\right)} + \frac{1}{x^{2}}}{3 \cos{\left(x \right)} - 4}$$

Simplify

The third derivative [src]

                                                           /                              2      \              
         /       2              \                          |       18*cos(x)        54*sin (x)   |              
         |  6*sin (x)           |                        3*|-1 + ------------- + ----------------|*log(x)*sin(x)
       9*|------------- + cos(x)|                          |     -4 + 3*cos(x)                  2|              
  2      \-4 + 3*cos(x)         /        9*sin(x)          \                     (-4 + 3*cos(x)) /              
- -- - -------------------------- + ------------------ - -------------------------------------------------------
   3       x*(-4 + 3*cos(x))         2                                        -4 + 3*cos(x)                     
  x                                 x *(-4 + 3*cos(x))                                                          
----------------------------------------------------------------------------------------------------------------
                                                 -4 + 3*cos(x)

$$\frac{- \frac{3 \left(-1 + \frac{18 \cos{\left(x \right)}}{3 \cos{\left(x \right)} - 4} + \frac{54 \sin^{2}{\left(x \right)}}{\left(3 \cos{\left(x \right)} - 4\right)^{2}}\right) \log{\left(x \right)} \sin{\left(x \right)}}{3 \cos{\left(x \right)} - 4} - \frac{9 \left(\cos{\left(x \right)} + \frac{6 \sin^{2}{\left(x \right)}}{3 \cos{\left(x \right)} - 4}\right)}{x \left(3 \cos{\left(x \right)} - 4\right)} + \frac{9 \sin{\left(x \right)}}{x^{2} \cdot \left(3 \cos{\left(x \right)} - 4\right)} - \frac{2}{x^{3}}}{3 \cos{\left(x \right)} - 4}$$

Simplify

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Derivative of (lnx)/(4-3cosx)

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