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Derivative of:
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Derivative of log(x)/(1-x)
Derivative of log(acot(x))
Integral of d{x}:
log(x)/(1-x)
Limit of the function:
log(x)/(1-x)
Identical expressions
log(x)/(one -x)
logarithm of (x) divide by (1 minus x)
logarithm of (x) divide by (one minus x)
logx/1-x
log(x) divide by (1-x)
Similar expressions
log(x)/(1+x)
sqrt(x)*log(x)/(1-x^2)

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

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

The solution

You have entered [src]

log(x)
------
1 - x

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

log(x)/(1 - x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
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: $-1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  2         6             3         6*log(x)
- -- - ----------- - ----------- + ---------
   3             2    2                    3
  x    x*(-1 + x)    x *(-1 + x)   (-1 + x) 
--------------------------------------------
                   -1 + x

$$\frac{\frac{6 \log{\left(x \right)}}{\left(x - 1\right)^{3}} - \frac{6}{x \left(x - 1\right)^{2}} - \frac{3}{x^{2} \left(x - 1\right)} - \frac{2}{x^{3}}}{x - 1}$$

Simplify

The graph

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Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution