Mister Exam

Other calculators

Derivative of arcsin(ln(x))/ln(arcsin(x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
asin(log(x))
------------
log(asin(x))
$$\frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)}}$$
asin(log(x))/log(asin(x))
The graph
The first derivative [src]
               1                             asin(log(x))          
------------------------------- - ---------------------------------
     _____________                   ________                      
    /        2                      /      2             2         
x*\/  1 - log (x) *log(asin(x))   \/  1 - x  *asin(x)*log (asin(x))
$$- \frac{\operatorname{asin}{\left(\log{\left(x \right)} \right)}}{\sqrt{1 - x^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)}^{2} \operatorname{asin}{\left(x \right)}} + \frac{1}{x \sqrt{1 - \log{\left(x \right)}^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)}}$$
The second derivative [src]
 /                      /     x                1                         2               \                                                                   \ 
 |         log(x)       |----------- + ----------------- + ------------------------------|*asin(log(x))                                                      | 
 |  1 + ------------    |        3/2   /      2\           /      2\                     |                                                                   | 
 |              2       |/     2\      \-1 + x /*asin(x)   \-1 + x /*asin(x)*log(asin(x))|                                                                   | 
 |      -1 + log (x)    \\1 - x /                                                        /                                         2                         | 
-|------------------- + ------------------------------------------------------------------------------- + ---------------------------------------------------| 
 |      _____________                                 asin(x)*log(asin(x))                                     ________    _____________                     | 
 | 2   /        2                                                                                             /      2    /        2                         | 
 \x *\/  1 - log (x)                                                                                      x*\/  1 - x  *\/  1 - log (x) *asin(x)*log(asin(x))/ 
---------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                          log(asin(x))                                                                         
$$- \frac{\frac{\left(\frac{x}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{1}{\left(x^{2} - 1\right) \operatorname{asin}{\left(x \right)}} + \frac{2}{\left(x^{2} - 1\right) \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}}\right) \operatorname{asin}{\left(\log{\left(x \right)} \right)}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{2}{x \sqrt{1 - x^{2}} \sqrt{1 - \log{\left(x \right)}^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}}{x^{2} \sqrt{1 - \log{\left(x \right)}^{2}}}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)}}$$
The third derivative [src]
                                          2         /                                            2                                                                                                                                   \                                                                                                                                           
         1           3*log(x)        3*log (x)      |     1                 2                 3*x              3*x                           6                                   6                                  6*x              |                  /     x                1                         2               \                                                       
2 - ------------ + ------------ + ---------------   |----------- + -------------------- + ----------- - ------------------ + --------------------------------- + ---------------------------------- - -------------------------------|*asin(log(x))   3*|----------- + ----------------- + ------------------------------|                     /       log(x)   \                
            2              2                    2   |        3/2           3/2                    5/2            2                   3/2                                 3/2                                   2                     |                  |        3/2   /      2\           /      2\                     |                   3*|1 + ------------|                
    -1 + log (x)   -1 + log (x)   /        2   \    |/     2\      /     2\        2      /     2\      /      2\            /     2\        2                   /     2\        2       2            /      2\                      |                  |/     2\      \-1 + x /*asin(x)   \-1 + x /*asin(x)*log(asin(x))|                     |            2   |                
                                  \-1 + log (x)/    \\1 - x /      \1 - x /   *asin (x)   \1 - x /      \-1 + x / *asin(x)   \1 - x /   *asin (x)*log(asin(x))   \1 - x /   *asin (x)*log (asin(x))   \-1 + x / *asin(x)*log(asin(x))/                  \\1 - x /                                                        /                     \    -1 + log (x)/                
------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------- + ----------------------------------------------------
                     _____________                                                                                                        asin(x)*log(asin(x))                                                                                                           _____________                                             ________    _____________                     
                3   /        2                                                                                                                                                                                                                                          /        2                                            2   /      2    /        2                         
               x *\/  1 - log (x)                                                                                                                                                                                                                                   x*\/  1 - log (x) *asin(x)*log(asin(x))                  x *\/  1 - x  *\/  1 - log (x) *asin(x)*log(asin(x))
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                   log(asin(x))                                                                                                                                                                                  
$$\frac{- \frac{\left(\frac{3 x^{2}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} - \frac{3 x}{\left(x^{2} - 1\right)^{2} \operatorname{asin}{\left(x \right)}} - \frac{6 x}{\left(x^{2} - 1\right)^{2} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{1}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{2}{\left(1 - x^{2}\right)^{\frac{3}{2}} \operatorname{asin}^{2}{\left(x \right)}} + \frac{6}{\left(1 - x^{2}\right)^{\frac{3}{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}^{2}{\left(x \right)}} + \frac{6}{\left(1 - x^{2}\right)^{\frac{3}{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)}^{2} \operatorname{asin}^{2}{\left(x \right)}}\right) \operatorname{asin}{\left(\log{\left(x \right)} \right)}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} - \frac{3 \left(\frac{x}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{1}{\left(x^{2} - 1\right) \operatorname{asin}{\left(x \right)}} + \frac{2}{\left(x^{2} - 1\right) \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}}\right)}{x \sqrt{1 - \log{\left(x \right)}^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{3 \left(1 + \frac{\log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1}\right)}{x^{2} \sqrt{1 - x^{2}} \sqrt{1 - \log{\left(x \right)}^{2}} \log{\left(\operatorname{asin}{\left(x \right)} \right)} \operatorname{asin}{\left(x \right)}} + \frac{2 + \frac{3 \log{\left(x \right)}}{\log{\left(x \right)}^{2} - 1} - \frac{1}{\log{\left(x \right)}^{2} - 1} + \frac{3 \log{\left(x \right)}^{2}}{\left(\log{\left(x \right)}^{2} - 1\right)^{2}}}{x^{3} \sqrt{1 - \log{\left(x \right)}^{2}}}}{\log{\left(\operatorname{asin}{\left(x \right)} \right)}}$$