Derivative of x*log(1-2x)

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x*log(1 - 2*x)

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

x*log(1 - 2*x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /       x    \
4*|1 - --------|
  \    -1 + 2*x/
----------------
    -1 + 2*x

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

Simplify

The third derivative [src]

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

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

Simplify