Derivative of y=(log5x)^arccosx

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   acos(x)     
log       (5*x)

\log{\left(5 x \right)}^{\operatorname{acos}{\left(x \right)}}

log(5*x)^acos(x)

Detail solution

Don't know the steps in finding this derivative.

But the derivative is

$\left(\log{\left(\operatorname{acos}{\left(x \right)} \right)} + 1\right) \operatorname{acos}^{\operatorname{acos}{\left(x \right)}}{\left(x \right)}$

The answer is:

\left(\log{\left(\operatorname{acos}{\left(x \right)} \right)} + 1\right) \operatorname{acos}^{\operatorname{acos}{\left(x \right)}}{\left(x \right)}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   acos(x)      /  log(log(5*x))    acos(x)  \
log       (5*x)*|- ------------- + ----------|
                |      ________    x*log(5*x)|
                |     /      2               |
                \   \/  1 - x                /

\left(- \frac{\log{\left(\log{\left(5 x \right)} \right)}}{\sqrt{1 - x^{2}}} + \frac{\operatorname{acos}{\left(x \right)}}{x \log{\left(5 x \right)}}\right) \log{\left(5 x \right)}^{\operatorname{acos}{\left(x \right)}}

Simplify

The second derivative [src]

                /                            2                                                                        \
   acos(x)      |/log(log(5*x))    acos(x)  \    x*log(log(5*x))     acos(x)       acos(x)                2           |
log       (5*x)*||------------- - ----------|  - --------------- - ----------- - ------------ - ----------------------|
                ||    ________    x*log(5*x)|              3/2      2             2    2             ________         |
                ||   /      2               |      /     2\        x *log(5*x)   x *log (5*x)       /      2          |
                \\ \/  1 - x                /      \1 - x /                                     x*\/  1 - x  *log(5*x)/

\left(- \frac{x \log{\left(\log{\left(5 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \left(\frac{\log{\left(\log{\left(5 x \right)} \right)}}{\sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x \log{\left(5 x \right)}}\right)^{2} - \frac{2}{x \sqrt{1 - x^{2}} \log{\left(5 x \right)}} - \frac{\operatorname{acos}{\left(x \right)}}{x^{2} \log{\left(5 x \right)}} - \frac{\operatorname{acos}{\left(x \right)}}{x^{2} \log{\left(5 x \right)}^{2}}\right) \log{\left(5 x \right)}^{\operatorname{acos}{\left(x \right)}}

Simplify

The third derivative [src]

                /                              3                                                                                                                                                      2                                                                                                               \
   acos(x)      |  /log(log(5*x))    acos(x)  \    log(log(5*x))            3               /log(log(5*x))    acos(x)  \ /x*log(log(5*x))     acos(x)       acos(x)                2           \   3*x *log(log(5*x))    2*acos(x)     2*acos(x)      3*acos(x)                3                         3            |
log       (5*x)*|- |------------- - ----------|  - ------------- - -------------------- + 3*|------------- - ----------|*|--------------- + ----------- + ------------ + ----------------------| - ------------------ + ----------- + ------------ + ------------ + ----------------------- + ------------------------|
                |  |    ________    x*log(5*x)|             3/2            3/2              |    ________    x*log(5*x)| |          3/2      2             2    2             ________         |              5/2        3             3    3         3    2              ________                  ________          |
                |  |   /      2               |     /     2\       /     2\                 |   /      2               | |  /     2\        x *log(5*x)   x *log (5*x)       /      2          |      /     2\          x *log(5*x)   x *log (5*x)   x *log (5*x)    2   /      2              2   /      2     2     |
                \  \ \/  1 - x                /     \1 - x /       \1 - x /   *log(5*x)     \ \/  1 - x                / \  \1 - x /                                     x*\/  1 - x  *log(5*x)/      \1 - x /                                                      x *\/  1 - x  *log(5*x)   x *\/  1 - x  *log (5*x)/

\left(- \frac{3 x^{2} \log{\left(\log{\left(5 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} - \left(\frac{\log{\left(\log{\left(5 x \right)} \right)}}{\sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x \log{\left(5 x \right)}}\right)^{3} + 3 \left(\frac{\log{\left(\log{\left(5 x \right)} \right)}}{\sqrt{1 - x^{2}}} - \frac{\operatorname{acos}{\left(x \right)}}{x \log{\left(5 x \right)}}\right) \left(\frac{x \log{\left(\log{\left(5 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{2}{x \sqrt{1 - x^{2}} \log{\left(5 x \right)}} + \frac{\operatorname{acos}{\left(x \right)}}{x^{2} \log{\left(5 x \right)}} + \frac{\operatorname{acos}{\left(x \right)}}{x^{2} \log{\left(5 x \right)}^{2}}\right) - \frac{\log{\left(\log{\left(5 x \right)} \right)}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{3}{\left(1 - x^{2}\right)^{\frac{3}{2}} \log{\left(5 x \right)}} + \frac{3}{x^{2} \sqrt{1 - x^{2}} \log{\left(5 x \right)}} + \frac{3}{x^{2} \sqrt{1 - x^{2}} \log{\left(5 x \right)}^{2}} + \frac{2 \operatorname{acos}{\left(x \right)}}{x^{3} \log{\left(5 x \right)}} + \frac{3 \operatorname{acos}{\left(x \right)}}{x^{3} \log{\left(5 x \right)}^{2}} + \frac{2 \operatorname{acos}{\left(x \right)}}{x^{3} \log{\left(5 x \right)}^{3}}\right) \log{\left(5 x \right)}^{\operatorname{acos}{\left(x \right)}}

Simplify

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Derivative of y=(log5x)^arccosx

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The solution