Derivative of -xlog2(x)-(1-x)log2(1-x)

The solution

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

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

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

Detail solution

Differentiate $- x \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{\log{\left(1 - x \right)}}{\log{\left(2 \right)}} \left(1 - x\right)$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = - x \log{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(2 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    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: $\log{\left(x \right)} + 1$
    So, the result is: $- \log{\left(x \right)} - 1$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of the constant $\log{\left(2 \right)}$ is zero.
  Now plug in to the quotient rule:
  
  $\frac{- \log{\left(x \right)} - 1}{\log{\left(2 \right)}}$
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. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = 1 - x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Differentiate $1 - x$ term by term:
        
        The derivative of the constant $1$ is zero.
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
        
        The result is: $-1$
      $g{\left(x \right)} = \log{\left(1 - x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Let $u = 1 - x$ .
      2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - x\right)$ :
        
        Differentiate $1 - x$ term by term:
        
        The derivative of the constant $1$ is zero.
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
        
        The result is: $-1$
        
        The result of the chain rule is:
        
        $- \frac{1}{1 - x}$
      The result is: $- \log{\left(1 - x \right)} - 1$
    So, the result is: $\frac{- \log{\left(1 - x \right)} - 1}{\log{\left(2 \right)}}$
  So, the result is: $- \frac{- \log{\left(1 - x \right)} - 1}{\log{\left(2 \right)}}$
The result is: $\frac{- \log{\left(x \right)} - 1}{\log{\left(2 \right)}} - \frac{- \log{\left(1 - x \right)} - 1}{\log{\left(2 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of -xlog2(x)-(1-x)log2(1-x)

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The solution