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The derivative of the function/ arccos((sqrt(x))/(1+x))

Derivative of arccos((sqrt(x))/(1+x))

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

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The solution

You have entered [src] 
    /  ___\
    |\/ x |
acos|-----|
    \1 + x/
acos⁡(xx+1)\operatorname{acos}{\left(\frac{\sqrt{x}}{x + 1} \right)} 
acos(sqrt(x)/(1 + x))
The graph
02468-8-6-4-2-10102.5-2.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 /                     ___  \ 
 |       1           \/ x   | 
-|--------------- - --------| 
 |    ___                  2| 
 \2*\/ x *(1 + x)   (1 + x) / 
------------------------------
          ______________      
         /        x           
        /  1 - --------       
       /              2       
     \/        (1 + x)
−−x(x+1)2+12x(x+1)−x(x+1)2+1- \frac{- \frac{\sqrt{x}}{\left(x + 1\right)^{2}} + \frac{1}{2 \sqrt{x} \left(x + 1\right)}}{\sqrt{- \frac{x}{\left(x + 1\right)^{2}} + 1}}
Simplify
The second derivative [src] 
                                                 /              ___\
                                    /      2*x \ |    1     2*\/ x |
                                    |-1 + -----|*|- ----- + -------|
                             ___    \     1 + x/ |    ___    1 + x |
  1            1         2*\/ x                  \  \/ x           /
------ + ------------- - -------- - --------------------------------
   3/2     ___                  2               2 /       x    \    
4*x      \/ x *(1 + x)   (1 + x)       4*(1 + x) *|1 - --------|    
                                                  |           2|    
                                                  \    (1 + x) /    
--------------------------------------------------------------------
                                 ______________                     
                                /        x                          
                    (1 + x)*   /  1 - --------                      
                              /              2                      
                            \/        (1 + x)
−2x(x+1)2−(2xx+1−1x)(2xx+1−1)4(x+1)2(−x(x+1)2+1)+1x(x+1)+14x32(x+1)−x(x+1)2+1\frac{- \frac{2 \sqrt{x}}{\left(x + 1\right)^{2}} - \frac{\left(\frac{2 \sqrt{x}}{x + 1} - \frac{1}{\sqrt{x}}\right) \left(\frac{2 x}{x + 1} - 1\right)}{4 \left(x + 1\right)^{2} \left(- \frac{x}{\left(x + 1\right)^{2}} + 1\right)} + \frac{1}{\sqrt{x} \left(x + 1\right)} + \frac{1}{4 x^{\frac{3}{2}}}}{\left(x + 1\right) \sqrt{- \frac{x}{\left(x + 1\right)^{2}} + 1}}
Simplify
The third derivative [src] 
                                                                     /              ___\                /           ___                 \                 2 /              ___\
                                                        /      3*x \ |    1     2*\/ x |   /      2*x \ | 1     8*\/ x           4      |     /      2*x \  |    1     2*\/ x |
                                                        |-2 + -----|*|- ----- + -------|   |-1 + -----|*|---- - -------- + -------------|   3*|-1 + -----| *|- ----- + -------|
                                ___                     \     1 + x/ |    ___    1 + x |   \     1 + x/ | 3/2          2     ___        |     \     1 + x/  |    ___    1 + x |
    3            3          6*\/ x           3                       \  \/ x           /                \x      (1 + x)    \/ x *(1 + x)/                   \  \/ x           /
- ------ - -------------- + -------- - -------------- + -------------------------------- - ---------------------------------------------- + -----------------------------------
     5/2     ___        2          3      3/2                       3 /       x    \                          2 /       x    \                                            2    
  8*x      \/ x *(1 + x)    (1 + x)    4*x   *(1 + x)      2*(1 + x) *|1 - --------|                 4*(1 + x) *|1 - --------|                            4 /       x    \     
                                                                      |           2|                            |           2|                   8*(1 + x) *|1 - --------|     
                                                                      \    (1 + x) /                            \    (1 + x) /                              |           2|     
                                                                                                                                                            \    (1 + x) /     
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                       ______________                                                                          
                                                                                      /        x                                                                               
                                                                          (1 + x)*   /  1 - --------                                                                           
                                                                                    /              2                                                                           
                                                                                  \/        (1 + x)
6x(x+1)3−(2xx+1−1)(−8x(x+1)2+4x(x+1)+1x32)4(x+1)2(−x(x+1)2+1)+(2xx+1−1x)(3xx+1−2)2(x+1)3(−x(x+1)2+1)+3(2xx+1−1x)(2xx+1−1)28(x+1)4(−x(x+1)2+1)2−3x(x+1)2−34x32(x+1)−38x52(x+1)−x(x+1)2+1\frac{\frac{6 \sqrt{x}}{\left(x + 1\right)^{3}} - \frac{\left(\frac{2 x}{x + 1} - 1\right) \left(- \frac{8 \sqrt{x}}{\left(x + 1\right)^{2}} + \frac{4}{\sqrt{x} \left(x + 1\right)} + \frac{1}{x^{\frac{3}{2}}}\right)}{4 \left(x + 1\right)^{2} \left(- \frac{x}{\left(x + 1\right)^{2}} + 1\right)} + \frac{\left(\frac{2 \sqrt{x}}{x + 1} - \frac{1}{\sqrt{x}}\right) \left(\frac{3 x}{x + 1} - 2\right)}{2 \left(x + 1\right)^{3} \left(- \frac{x}{\left(x + 1\right)^{2}} + 1\right)} + \frac{3 \left(\frac{2 \sqrt{x}}{x + 1} - \frac{1}{\sqrt{x}}\right) \left(\frac{2 x}{x + 1} - 1\right)^{2}}{8 \left(x + 1\right)^{4} \left(- \frac{x}{\left(x + 1\right)^{2}} + 1\right)^{2}} - \frac{3}{\sqrt{x} \left(x + 1\right)^{2}} - \frac{3}{4 x^{\frac{3}{2}} \left(x + 1\right)} - \frac{3}{8 x^{\frac{5}{2}}}}{\left(x + 1\right) \sqrt{- \frac{x}{\left(x + 1\right)^{2}} + 1}}
Simplify
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