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

Derivative of x(ln)(1-x)

The solution

You have entered [src]

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

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

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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 $1$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\log{\left(x \right)} + 1$
$g{\left(x \right)} = 1 - x$ ; 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: $- x \log{\left(x \right)} + \left(1 - x\right) \left(\log{\left(x \right)} + 1\right)$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     -1 + x
-3 + ------
       x   
-----------
     x

$$\frac{-3 + \frac{x - 1}{x}}{x}$$

Simplify