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Derivative of logx/(1+2*log)*sin(x)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   log(x)          
------------*sin(x)
1 + 2*log(x)       
$$\frac{\log{\left(x \right)}}{2 \log{\left(x \right)} + 1} \sin{\left(x \right)}$$
(log(x)/(1 + 2*log(x)))*sin(x)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Apply the product rule:

      ; to find :

      1. The derivative of is .

      ; to find :

      1. The derivative of sine is cosine:

      The result is:

    To find :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. The derivative of is .

        So, the result is:

      The result is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
/       1                2*log(x)    \          cos(x)*log(x)
|---------------- - -----------------|*sin(x) + -------------
|x*(1 + 2*log(x))                   2|           1 + 2*log(x)
\                   x*(1 + 2*log(x)) /                       
$$\left(\frac{1}{x \left(2 \log{\left(x \right)} + 1\right)} - \frac{2 \log{\left(x \right)}}{x \left(2 \log{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)} + \frac{\log{\left(x \right)} \cos{\left(x \right)}}{2 \log{\left(x \right)} + 1}$$
The second derivative [src]
                 /                     /         4      \       \                                     
                 |                   2*|1 + ------------|*log(x)|                                     
                 |         4           \    1 + 2*log(x)/       |            /      2*log(x)  \       
                 |1 + ------------ - ---------------------------|*sin(x)   2*|1 - ------------|*cos(x)
                 \    1 + 2*log(x)           1 + 2*log(x)       /            \    1 + 2*log(x)/       
-log(x)*sin(x) - ------------------------------------------------------- + ---------------------------
                                             2                                          x             
                                            x                                                         
------------------------------------------------------------------------------------------------------
                                             1 + 2*log(x)                                             
$$\frac{- \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \left(1 - \frac{2 \log{\left(x \right)}}{2 \log{\left(x \right)} + 1}\right) \cos{\left(x \right)}}{x} - \frac{\left(- \frac{2 \left(1 + \frac{4}{2 \log{\left(x \right)} + 1}\right) \log{\left(x \right)}}{2 \log{\left(x \right)} + 1} + 1 + \frac{4}{2 \log{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{x^{2}}}{2 \log{\left(x \right)} + 1}$$
The third derivative [src]
                                                                                                             /                                            /         6                12      \       \       
                                                 /                     /         4      \       \            |                     /         4      \   2*|1 + ------------ + ---------------|*log(x)|       
                                                 |                   2*|1 + ------------|*log(x)|            |                   3*|1 + ------------|     |    1 + 2*log(x)                 2|       |       
                   /      2*log(x)  \            |         4           \    1 + 2*log(x)/       |            |         3           \    1 + 2*log(x)/     \                   (1 + 2*log(x)) /       |       
                 3*|1 - ------------|*sin(x)   3*|1 + ------------ - ---------------------------|*cos(x)   2*|1 + ------------ + -------------------- - ---------------------------------------------|*sin(x)
                   \    1 + 2*log(x)/            \    1 + 2*log(x)           1 + 2*log(x)       /            \    1 + 2*log(x)       1 + 2*log(x)                        1 + 2*log(x)                /       
-cos(x)*log(x) - --------------------------- - --------------------------------------------------------- + --------------------------------------------------------------------------------------------------
                              x                                             2                                                                               3                                                
                                                                           x                                                                               x                                                 
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                 1 + 2*log(x)                                                                                                
$$\frac{- \log{\left(x \right)} \cos{\left(x \right)} - \frac{3 \left(1 - \frac{2 \log{\left(x \right)}}{2 \log{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{x} - \frac{3 \left(- \frac{2 \left(1 + \frac{4}{2 \log{\left(x \right)} + 1}\right) \log{\left(x \right)}}{2 \log{\left(x \right)} + 1} + 1 + \frac{4}{2 \log{\left(x \right)} + 1}\right) \cos{\left(x \right)}}{x^{2}} + \frac{2 \left(\frac{3 \left(1 + \frac{4}{2 \log{\left(x \right)} + 1}\right)}{2 \log{\left(x \right)} + 1} + 1 - \frac{2 \left(1 + \frac{6}{2 \log{\left(x \right)} + 1} + \frac{12}{\left(2 \log{\left(x \right)} + 1\right)^{2}}\right) \log{\left(x \right)}}{2 \log{\left(x \right)} + 1} + \frac{3}{2 \log{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{x^{3}}}{2 \log{\left(x \right)} + 1}$$