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u-r+ two /(k- one)*(one -(p/r)**((k- one)/(two *k)))*с
u minus r plus 2 divide by (k minus 1) multiply by (1 minus (p divide by r) multiply by multiply by ((k minus 1) divide by (2 multiply by k))) multiply by с
u minus r plus two divide by (k minus one) multiply by (one minus (p divide by r) multiply by multiply by ((k minus one) divide by (two multiply by k))) multiply by с
u-r+2/(k-1)(1-(p/r)((k-1)/(2k)))с
u-r+2/k-11-p/rk-1/2kс
u-r+2 divide by (k-1)*(1-(p divide by r)**((k-1) divide by (2*k)))*с
Similar expressions
u-r+2/(k-1)*(1-(p/r)**((k+1)/(2*k)))*с
u+r+2/(k-1)*(1-(p/r)**((k-1)/(2*k)))*с
u-r+2/(k-1)*(1+(p/r)**((k-1)/(2*k)))*с
u-r-2/(k-1)*(1-(p/r)**((k-1)/(2*k)))*с
u-r+2/(k+1)*(1-(p/r)**((k-1)/(2*k)))*с

The derivative of the function/ u-r+2/(k-1)*(1-(p/r)**((k-1)/(2*k)))*с

Derivative of u-r+2/(k-1)*(1-(p/r)**((k-1)/(2k)))с

The solution

You have entered [src]

              /       k - 1\  
              |       -----|  
              |        2*k |  
          2   |    /p\     |  
u - r + -----*|1 - |-|     |*c
        k - 1 \    \r/     /

$$c \left(1 - \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}}\right) \frac{2}{k - 1} + \left(- r + u\right)$$

u - r + ((2/(k - 1))*(1 - (p/r)^((k - 1)/((2*k)))))*c

Detail solution

Differentiate $c \left(1 - \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}}\right) \frac{2}{k - 1} + \left(- r + u\right)$ term by term:
1. The derivative of the constant $- r + u$ is zero.
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Differentiate $1 - \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}}$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Let $u = \frac{p}{r}$ .
        
        Apply the power rule: $u^{\frac{k - 1}{2 k}}$ goes to $\frac{\frac{1}{2 k} u^{\frac{k - 1}{2 k}} \left(k - 1\right)}{u}$
        
        Then, apply the chain rule. Multiply by $\frac{\partial}{\partial p} \frac{p}{r}$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $p$ goes to $1$
        
        So, the result is: $\frac{1}{r}$
        
        The result of the chain rule is:
        
        $\frac{\left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}} \left(k - 1\right)}{2 k p}$
        
        So, the result is: $- \frac{\left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}} \left(k - 1\right)}{2 k p}$
      The result is: $- \frac{\left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}} \left(k - 1\right)}{2 k p}$
    So, the result is: $- \frac{\left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}}}{k p}$
  So, the result is: $- \frac{c \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}}}{k p}$
The result is: $- \frac{c \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}}}{k p}$
Now simplify:

$- \frac{c \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}}}{k p}$

The answer is:

$- \frac{c \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}}}{k p}$

The first derivative [src]

      k - 1 
      ----- 
       2*k  
   /p\      
-c*|-|      
   \r/      
------------
    k*p

$$- \frac{c \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}}}{k p}$$

Simplify

The second derivative [src]

     -1 + k             
     ------             
      2*k               
  /p\       /    -1 + k\
c*|-|      *|1 - ------|
  \r/       \     2*k  /
------------------------
             2          
          k*p

$$\frac{c \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}} \left(1 - \frac{k - 1}{2 k}\right)}{k p^{2}}$$

Simplify

The third derivative [src]

     -1 + k                              
     ------                              
      2*k   /             2             \
  /p\       |     (-1 + k)    3*(-1 + k)|
c*|-|      *|-2 - --------- + ----------|
  \r/       |           2        2*k    |
            \        4*k                /
-----------------------------------------
                      3                  
                   k*p

$$\frac{c \left(\frac{p}{r}\right)^{\frac{k - 1}{2 k}} \left(-2 + \frac{3 \left(k - 1\right)}{2 k} - \frac{\left(k - 1\right)^{2}}{4 k^{2}}\right)}{k p^{3}}$$

Simplify

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Derivative of u-r+2/(k-1)*(1-(p/r)**((k-1)/(2k)))с

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The solution

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Derivative of u-r+2/(k-1)*(1-(p/r)**((k-1)/(2*k)))*с

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Piecewise:

The solution

Derivative of u-r+2/(k-1)*(1-(p/r)**((k-1)/(2k)))с