Mister Exam

Other calculators

The derivative of the function/ (log(arcsin(sqrt(x)/(x-5))))log3

Derivative of (log(arcsin(sqrt(x)/(x-5))))log3

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /    /  ___\\       
   |    |\/ x ||       
log|asin|-----||*log(3)
   \    \x - 5//
log⁡(3)log⁡(asin⁡(xx−5))\log{\left(3 \right)} \log{\left(\operatorname{asin}{\left(\frac{\sqrt{x}}{x - 5} \right)} \right)} 
log(asin(sqrt(x)/(x - 5)))*log(3)
The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/                     ___  \       
|       1           \/ x   |       
|--------------- - --------|*log(3)
|    ___                  2|       
\2*\/ x *(x - 5)   (x - 5) /       
-----------------------------------
       ______________     /  ___\  
      /        x          |\/ x |  
     /  1 - -------- *asin|-----|  
    /              2      \x - 5/  
  \/        (x - 5)
(−x(x−5)2+12x(x−5))log⁡(3)−x(x−5)2+1asin⁡(xx−5)\frac{\left(- \frac{\sqrt{x}}{\left(x - 5\right)^{2}} + \frac{1}{2 \sqrt{x} \left(x - 5\right)}\right) \log{\left(3 \right)}}{\sqrt{- \frac{x}{\left(x - 5\right)^{2}} + 1} \operatorname{asin}{\left(\frac{\sqrt{x}}{x - 5} \right)}}
Simplify
The second derivative [src] 
/                                                                                                      2         \       
|              ___                                   /              ___\            /              ___\          |       
|   1      8*\/ x           4          /      2*x  \ |    1     2*\/ x |            |    1     2*\/ x |          |       
|  ---- - --------- + --------------   |-1 + ------|*|- ----- + -------|            |- ----- + -------|          |       
|   3/2           2     ___            \     -5 + x/ |    ___    -5 + x|            |    ___    -5 + x|          |       
|  x      (-5 + x)    \/ x *(-5 + x)                 \  \/ x           /            \  \/ x           /          |       
|- --------------------------------- + --------------------------------- + --------------------------------------|*log(3)
|              _______________                           3/2                                             /  ___ \|       
|             /         x                 /        x    \            2     /         x    \              |\/ x  ||       
|            /  1 - ---------             |1 - ---------|   *(-5 + x)      |-1 + ---------|*(-5 + x)*asin|------||       
|           /               2             |            2|                  |             2|              \-5 + x/|       
\         \/        (-5 + x)              \    (-5 + x) /                  \     (-5 + x) /                      /       
-------------------------------------------------------------------------------------------------------------------------
                                                                /  ___ \                                                 
                                                                |\/ x  |                                                 
                                                 4*(-5 + x)*asin|------|                                                 
                                                                \-5 + x/
(−−8x(x−5)2+4x(x−5)+1x32−x(x−5)2+1+(2xx−5−1x)2(x−5)(x(x−5)2−1)asin⁡(xx−5)+(2xx−5−1x)(2xx−5−1)(x−5)2(−x(x−5)2+1)32)log⁡(3)4(x−5)asin⁡(xx−5)\frac{\left(- \frac{- \frac{8 \sqrt{x}}{\left(x - 5\right)^{2}} + \frac{4}{\sqrt{x} \left(x - 5\right)} + \frac{1}{x^{\frac{3}{2}}}}{\sqrt{- \frac{x}{\left(x - 5\right)^{2}} + 1}} + \frac{\left(\frac{2 \sqrt{x}}{x - 5} - \frac{1}{\sqrt{x}}\right)^{2}}{\left(x - 5\right) \left(\frac{x}{\left(x - 5\right)^{2}} - 1\right) \operatorname{asin}{\left(\frac{\sqrt{x}}{x - 5} \right)}} + \frac{\left(\frac{2 \sqrt{x}}{x - 5} - \frac{1}{\sqrt{x}}\right) \left(\frac{2 x}{x - 5} - 1\right)}{\left(x - 5\right)^{2} \left(- \frac{x}{\left(x - 5\right)^{2}} + 1\right)^{\frac{3}{2}}}\right) \log{\left(3 \right)}}{4 \left(x - 5\right) \operatorname{asin}{\left(\frac{\sqrt{x}}{x - 5} \right)}}
Simplify
The third derivative [src] 
/                                                                                                                                                                     3                                                                                                                                                      2                \       
|  /             ___                                  \                   /              ___\                  2 /              ___\               /              ___\                              /            ___                  \     /              ___\ /            ___                  \       /              ___\                 |       
|  | 1      16*\/ x          2                8       |     /      3*x  \ |    1     2*\/ x |     /      2*x  \  |    1     2*\/ x |               |    1     2*\/ x |                /      2*x  \ | 1      8*\/ x           4       |     |    1     2*\/ x | | 1      8*\/ x           4       |       |    1     2*\/ x |  /      2*x  \  |       
|3*|---- - --------- + ------------- + ---------------|   4*|-2 + ------|*|- ----- + -------|   3*|-1 + ------| *|- ----- + -------|             2*|- ----- + -------|              2*|-1 + ------|*|---- - --------- + --------------|   3*|- ----- + -------|*|---- - --------- + --------------|     3*|- ----- + -------| *|-1 + ------|  |       
|  | 5/2           3    3/2              ___         2|     \     -5 + x/ |    ___    -5 + x|     \     -5 + x/  |    ___    -5 + x|               |    ___    -5 + x|                \     -5 + x/ | 3/2           2     ___         |     |    ___    -5 + x| | 3/2           2     ___         |       |    ___    -5 + x|  \     -5 + x/  |       
|  \x      (-5 + x)    x   *(-5 + x)   \/ x *(-5 + x) /                   \  \/ x           /                    \  \/ x           /               \  \/ x           /                              \x      (-5 + x)    \/ x *(-5 + x)/     \  \/ x           / \x      (-5 + x)    \/ x *(-5 + x)/       \  \/ x           /                 |       
|------------------------------------------------------ - ----------------------------------- - ------------------------------------ - ------------------------------------------ + --------------------------------------------------- + --------------------------------------------------------- + ----------------------------------------|*log(3)
|                      _______________                                       3/2                                   5/2                                3/2                /  ___ \                              3/2                                                                /  ___ \                            2               /  ___ \|       
|                     /         x                             /        x    \            3          /        x    \            4       /        x    \            2     2|\/ x  |               /        x    \            2                        /         x    \              |\/ x  |            /         x    \          3     |\/ x  ||       
|                    /  1 - ---------                         |1 - ---------|   *(-5 + x)           |1 - ---------|   *(-5 + x)        |1 - ---------|   *(-5 + x) *asin |------|               |1 - ---------|   *(-5 + x)                         |-1 + ---------|*(-5 + x)*asin|------|            |-1 + ---------| *(-5 + x) *asin|------||       
|                   /               2                         |            2|                       |            2|                    |            2|                   \-5 + x/               |            2|                                     |             2|              \-5 + x/            |             2|                \-5 + x/|       
\                 \/        (-5 + x)                          \    (-5 + x) /                       \    (-5 + x) /                    \    (-5 + x) /                                          \    (-5 + x) /                                     \     (-5 + x) /                                  \     (-5 + x) /                        /       
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                              /  ___ \                                                                                                                                                                
                                                                                                                                                                              |\/ x  |                                                                                                                                                                
                                                                                                                                                               8*(-5 + x)*asin|------|                                                                                                                                                                
                                                                                                                                                                              \-5 + x/
(3(−16x(x−5)3+8x(x−5)2+2x32(x−5)+1x52)−x(x−5)2+1+3(2xx−5−1x)(−8x(x−5)2+4x(x−5)+1x32)(x−5)(x(x−5)2−1)asin⁡(xx−5)−2(2xx−5−1x)3(x−5)2(−x(x−5)2+1)32asin⁡2(xx−5)+2(2xx−5−1)(−8x(x−5)2+4x(x−5)+1x32)(x−5)2(−x(x−5)2+1)32+3(2xx−5−1x)2(2xx−5−1)(x−5)3(x(x−5)2−1)2asin⁡(xx−5)−4(2xx−5−1x)(3xx−5−2)(x−5)3(−x(x−5)2+1)32−3(2xx−5−1x)(2xx−5−1)2(x−5)4(−x(x−5)2+1)52)log⁡(3)8(x−5)asin⁡(xx−5)\frac{\left(\frac{3 \left(- \frac{16 \sqrt{x}}{\left(x - 5\right)^{3}} + \frac{8}{\sqrt{x} \left(x - 5\right)^{2}} + \frac{2}{x^{\frac{3}{2}} \left(x - 5\right)} + \frac{1}{x^{\frac{5}{2}}}\right)}{\sqrt{- \frac{x}{\left(x - 5\right)^{2}} + 1}} + \frac{3 \left(\frac{2 \sqrt{x}}{x - 5} - \frac{1}{\sqrt{x}}\right) \left(- \frac{8 \sqrt{x}}{\left(x - 5\right)^{2}} + \frac{4}{\sqrt{x} \left(x - 5\right)} + \frac{1}{x^{\frac{3}{2}}}\right)}{\left(x - 5\right) \left(\frac{x}{\left(x - 5\right)^{2}} - 1\right) \operatorname{asin}{\left(\frac{\sqrt{x}}{x - 5} \right)}} - \frac{2 \left(\frac{2 \sqrt{x}}{x - 5} - \frac{1}{\sqrt{x}}\right)^{3}}{\left(x - 5\right)^{2} \left(- \frac{x}{\left(x - 5\right)^{2}} + 1\right)^{\frac{3}{2}} \operatorname{asin}^{2}{\left(\frac{\sqrt{x}}{x - 5} \right)}} + \frac{2 \left(\frac{2 x}{x - 5} - 1\right) \left(- \frac{8 \sqrt{x}}{\left(x - 5\right)^{2}} + \frac{4}{\sqrt{x} \left(x - 5\right)} + \frac{1}{x^{\frac{3}{2}}}\right)}{\left(x - 5\right)^{2} \left(- \frac{x}{\left(x - 5\right)^{2}} + 1\right)^{\frac{3}{2}}} + \frac{3 \left(\frac{2 \sqrt{x}}{x - 5} - \frac{1}{\sqrt{x}}\right)^{2} \left(\frac{2 x}{x - 5} - 1\right)}{\left(x - 5\right)^{3} \left(\frac{x}{\left(x - 5\right)^{2}} - 1\right)^{2} \operatorname{asin}{\left(\frac{\sqrt{x}}{x - 5} \right)}} - \frac{4 \left(\frac{2 \sqrt{x}}{x - 5} - \frac{1}{\sqrt{x}}\right) \left(\frac{3 x}{x - 5} - 2\right)}{\left(x - 5\right)^{3} \left(- \frac{x}{\left(x - 5\right)^{2}} + 1\right)^{\frac{3}{2}}} - \frac{3 \left(\frac{2 \sqrt{x}}{x - 5} - \frac{1}{\sqrt{x}}\right) \left(\frac{2 x}{x - 5} - 1\right)^{2}}{\left(x - 5\right)^{4} \left(- \frac{x}{\left(x - 5\right)^{2}} + 1\right)^{\frac{5}{2}}}\right) \log{\left(3 \right)}}{8 \left(x - 5\right) \operatorname{asin}{\left(\frac{\sqrt{x}}{x - 5} \right)}}
Simplify
    © Mister Exam – Calculator