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Similar expressions
ln^2*x/sinx

The derivative of the function/ ln^2*x/sin(x)

Derivative of ln^2*x/sin(x)

The solution

You have entered [src]

   2   
log (x)
-------
 sin(x)

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

log(x)^2/sin(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $\frac{2 \log{\left(x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     2                     
  log (x)*cos(x)   2*log(x)
- -------------- + --------
        2          x*sin(x)
     sin (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                      /         2   \                  /         2   \                                
                      |    2*cos (x)|             2    |    6*cos (x)|                                
                    6*|1 + ---------|*log(x)   log (x)*|5 + ---------|*cos(x)                         
                      |        2    |                  |        2    |                                
2*(-3 + 2*log(x))     \     sin (x) /                  \     sin (x) /          6*(-1 + log(x))*cos(x)
----------------- + ------------------------ - ------------------------------ + ----------------------
         3                     x                           sin(x)                      2              
        x                                                                             x *sin(x)       
------------------------------------------------------------------------------------------------------
                                                sin(x)

$$\frac{- \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(x \right)}^{2} \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{6 \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(x \right)}}{x} + \frac{6 \left(\log{\left(x \right)} - 1\right) \cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} + \frac{2 \left(2 \log{\left(x \right)} - 3\right)}{x^{3}}}{\sin{\left(x \right)}}$$

Simplify

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Derivative of ln^2*x/sin(x)

The graph:

Piecewise:

The solution