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The derivative of the function/ lnx^(arcsinx)

Derivative of lnx^(arcsinx)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   asin(x)   
log       (x)
$$\log{\left(x \right)}^{\operatorname{asin}{\left(x \right)}}$$ 
log(x)^asin(x)
Detail solution

  1. Don't know the steps in finding this derivative.

    But the derivative is

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   asin(x)    /log(log(x))   asin(x) \
log       (x)*|----------- + --------|
              |   ________   x*log(x)|
              |  /      2            |
              \\/  1 - x             /
$$\left(\frac{\log{\left(\log{\left(x \right)} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x \log{\left(x \right)}}\right) \log{\left(x \right)}^{\operatorname{asin}{\left(x \right)}}$$
Simplify
The second derivative [src] 
              /                        2                                                                \
   asin(x)    |/log(log(x))   asin(x) \    x*log(log(x))    asin(x)     asin(x)              2          |
log       (x)*||----------- + --------|  + ------------- - --------- - ---------- + --------------------|
              ||   ________   x*log(x)|             3/2     2           2    2           ________       |
              ||  /      2            |     /     2\       x *log(x)   x *log (x)       /      2        |
              \\\/  1 - x             /     \1 - x /                                x*\/  1 - x  *log(x)/
$$\left(\frac{x \log{\left(\log{\left(x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \left(\frac{\log{\left(\log{\left(x \right)} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x \log{\left(x \right)}}\right)^{2} + \frac{2}{x \sqrt{1 - x^{2}} \log{\left(x \right)}} - \frac{\operatorname{asin}{\left(x \right)}}{x^{2} \log{\left(x \right)}} - \frac{\operatorname{asin}{\left(x \right)}}{x^{2} \log{\left(x \right)}^{2}}\right) \log{\left(x \right)}^{\operatorname{asin}{\left(x \right)}}$$
Simplify
The third derivative [src] 
              /                        3                                                                                                                                                                                                                             2            \
   asin(x)    |/log(log(x))   asin(x) \    log(log(x))           3              /log(log(x))   asin(x) \ /x*log(log(x))    asin(x)     asin(x)              2          \             3                       3              2*asin(x)   2*asin(x)    3*asin(x)    3*x *log(log(x))|
log       (x)*||----------- + --------|  + ----------- + ------------------ + 3*|----------- + --------|*|------------- - --------- - ---------- + --------------------| - --------------------- - ---------------------- + --------- + ---------- + ---------- + ----------------|
              ||   ________   x*log(x)|            3/2           3/2            |   ________   x*log(x)| |         3/2     2           2    2           ________       |         ________                ________            3           3    3       3    2                5/2   |
              ||  /      2            |    /     2\      /     2\               |  /      2            | | /     2\       x *log(x)   x *log (x)       /      2        |    2   /      2            2   /      2     2      x *log(x)   x *log (x)   x *log (x)     /     2\      |
              \\\/  1 - x             /    \1 - x /      \1 - x /   *log(x)     \\/  1 - x             / \ \1 - x /                                x*\/  1 - x  *log(x)/   x *\/  1 - x  *log(x)   x *\/  1 - x  *log (x)                                           \1 - x /      /
$$\left(\frac{3 x^{2} \log{\left(\log{\left(x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} + \left(\frac{\log{\left(\log{\left(x \right)} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x \log{\left(x \right)}}\right)^{3} + 3 \left(\frac{\log{\left(\log{\left(x \right)} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{asin}{\left(x \right)}}{x \log{\left(x \right)}}\right) \left(\frac{x \log{\left(\log{\left(x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{2}{x \sqrt{1 - x^{2}} \log{\left(x \right)}} - \frac{\operatorname{asin}{\left(x \right)}}{x^{2} \log{\left(x \right)}} - \frac{\operatorname{asin}{\left(x \right)}}{x^{2} \log{\left(x \right)}^{2}}\right) + \frac{\log{\left(\log{\left(x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{3}{\left(1 - x^{2}\right)^{\frac{3}{2}} \log{\left(x \right)}} - \frac{3}{x^{2} \sqrt{1 - x^{2}} \log{\left(x \right)}} - \frac{3}{x^{2} \sqrt{1 - x^{2}} \log{\left(x \right)}^{2}} + \frac{2 \operatorname{asin}{\left(x \right)}}{x^{3} \log{\left(x \right)}} + \frac{3 \operatorname{asin}{\left(x \right)}}{x^{3} \log{\left(x \right)}^{2}} + \frac{2 \operatorname{asin}{\left(x \right)}}{x^{3} \log{\left(x \right)}^{3}}\right) \log{\left(x \right)}^{\operatorname{asin}{\left(x \right)}}$$
Simplify
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