Derivative of 1/(x(1-x))

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    1    
---------
x*(1 - x)

$$\frac{1}{x \left(1 - x\right)}$$

1/(x*(1 - x))

Detail solution

Let $u = x \left(1 - x\right)$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} x \left(1 - x\right)$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = 1 - x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $1 - x$ 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.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $-1$
    The result is: $-1$
  The result is: $1 - 2 x$
The result of the chain rule is:

$- \frac{1 - 2 x}{x^{2} \left(1 - x\right)^{2}}$
Now simplify:

$\frac{2 x - 1}{x^{2} \left(x - 1\right)^{2}}$

The answer is:

$\frac{2 x - 1}{x^{2} \left(x - 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1               
---------*(-1 + 2*x)
x*(1 - x)           
--------------------
     x*(1 - x)

$$\frac{\frac{1}{x \left(1 - x\right)} \left(2 x - 1\right)}{x \left(1 - x\right)}$$

Simplify

The second derivative [src]

    -1 + 2*x   -1 + 2*x              /1     1   \
2 - -------- - -------- - (-1 + 2*x)*|- + ------|
       x        -1 + x               \x   -1 + x/
-------------------------------------------------
                    2         2                  
                   x *(-1 + x)

$$\frac{- \left(2 x - 1\right) \left(\frac{1}{x - 1} + \frac{1}{x}\right) + 2 - \frac{2 x - 1}{x - 1} - \frac{2 x - 1}{x}}{x^{2} \left(x - 1\right)^{2}}$$

Simplify

The third derivative [src]

                                                                                                     /1     1   \              /1     1   \               
                                                                                          (-1 + 2*x)*|- + ------|   (-1 + 2*x)*|- + ------|               
  8     8                   /1        1           1     \   3*(-1 + 2*x)   3*(-1 + 2*x)              \x   -1 + x/              \x   -1 + x/   4*(-1 + 2*x)
- - - ------ + 2*(-1 + 2*x)*|-- + --------- + ----------| + ------------ + ------------ + ----------------------- + ----------------------- + ------------
  x   -1 + x                | 2           2   x*(-1 + x)|         2                 2                x                       -1 + x            x*(-1 + x) 
                            \x    (-1 + x)              /        x          (-1 + x)                                                                      
----------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                        2         2                                                                       
                                                                       x *(-1 + x)

$$\frac{2 \left(2 x - 1\right) \left(\frac{1}{\left(x - 1\right)^{2}} + \frac{1}{x \left(x - 1\right)} + \frac{1}{x^{2}}\right) + \frac{\left(2 x - 1\right) \left(\frac{1}{x - 1} + \frac{1}{x}\right)}{x - 1} - \frac{8}{x - 1} + \frac{3 \left(2 x - 1\right)}{\left(x - 1\right)^{2}} + \frac{\left(2 x - 1\right) \left(\frac{1}{x - 1} + \frac{1}{x}\right)}{x} - \frac{8}{x} + \frac{4 \left(2 x - 1\right)}{x \left(x - 1\right)} + \frac{3 \left(2 x - 1\right)}{x^{2}}}{x^{2} \left(x - 1\right)^{2}}$$

Simplify

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Derivative of 1/(x(1-x))

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The solution