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The derivative of the function/ x/(1-x)

Derivative of x/(1-x)

The solution

You have entered [src]

  x  
-----
1 - x

$$\frac{x}{1 - x}$$

x/(1 - x)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x$ and $g{\left(x \right)} = 1 - x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
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: $-1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       x   \
6*|-1 + ------|
  \     -1 + x/
---------------
           3   
   (-1 + x)

$$\frac{6 \left(\frac{x}{x - 1} - 1\right)}{\left(x - 1\right)^{3}}$$

Simplify

The graph

Derivative of x/(1-x)