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Identical expressions
x*ln(one - two x^2)
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x multiply by ln(one minus two x squared )
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Similar expressions
x*ln(1+2x^2)

The derivative of the function/ x*ln(1-2x^2)

Derivative of x*ln(1-2x^2)

The solution

You have entered [src]

     /       2\
x*log\1 - 2*x /

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

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

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 $1$
$g{\left(x \right)} = \log{\left(1 - 2 x^{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 1 - 2 x^{2}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - 2 x^{2}\right)$ :
  1. Differentiate $1 - 2 x^{2}$ 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: $x^{2}$ goes to $2 x$
      So, the result is: $- 4 x$
    The result is: $- 4 x$
  The result of the chain rule is:
  
  $- \frac{4 x}{1 - 2 x^{2}}$
The result is: $- \frac{4 x^{2}}{1 - 2 x^{2}} + \log{\left(1 - 2 x^{2} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

5-я производная [src]

   /                                     /          2            4    \\
   |                                   2 |      40*x         64*x     ||
   |                                8*x *|5 - --------- + ------------||
   |                                     |            2              2||
   |             4            2          |    -1 + 2*x    /        2\ ||
   |        160*x         80*x           \                \-1 + 2*x / /|
48*|-5 - ------------ + --------- + -----------------------------------|
   |                2           2                        2             |
   |     /        2\    -1 + 2*x                 -1 + 2*x              |
   \     \-1 + 2*x /                                                   /
------------------------------------------------------------------------
                                         2                              
                              /        2\                               
                              \-1 + 2*x /

$$\frac{48 \left(- \frac{160 x^{4}}{\left(2 x^{2} - 1\right)^{2}} + \frac{8 x^{2} \left(\frac{64 x^{4}}{\left(2 x^{2} - 1\right)^{2}} - \frac{40 x^{2}}{2 x^{2} - 1} + 5\right)}{2 x^{2} - 1} + \frac{80 x^{2}}{2 x^{2} - 1} - 5\right)}{\left(2 x^{2} - 1\right)^{2}}$$

Simplify

The third derivative [src]

  /                     /           2  \\
  |                   2 |        8*x   ||
  |                4*x *|-3 + ---------||
  |          2          |             2||
  |      12*x           \     -1 + 2*x /|
4*|3 - --------- + ---------------------|
  |            2                 2      |
  \    -1 + 2*x          -1 + 2*x       /
-----------------------------------------
                        2                
                -1 + 2*x

$$\frac{4 \left(\frac{4 x^{2} \left(\frac{8 x^{2}}{2 x^{2} - 1} - 3\right)}{2 x^{2} - 1} - \frac{12 x^{2}}{2 x^{2} - 1} + 3\right)}{2 x^{2} - 1}$$

Simplify

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Derivative of x*ln(1-2x^2)

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Piecewise:

The solution