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logx/(one + two *log)*sin(x)
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logx/(1+2log)sin(x)
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Similar expressions
logx/(1-2*log)*sin(x)
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The derivative of the function/ logx/(1+2*log)*sin(x)

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

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

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)}$ and $g{\left(x \right)} = 2 \log{\left(x \right)} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $2 \log{\left(x \right)} + 1$ 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. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    So, the result is: $\frac{2}{x}$
  The result is: $\frac{2}{x}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

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}

Simplify

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}

Simplify

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}

Simplify

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

The graph:

Piecewise:

The solution

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

The graph:

Piecewise:

The solution

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