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ln(one +x^ two)*ln(one -2x)
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Similar expressions
ln(1-x^2)*ln(1-2x)
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The derivative of the function/ ln(1+x^2)*ln(1-2x)

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

The solution

You have entered [src]

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

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

d /   /     2\             \
--\log\1 + x /*log(1 - 2*x)/
dx

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

Transform

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                           /         2 \       /         2 \             \
  |                                           |      2*x  |       |      4*x  |             |
  |                                         3*|-1 + ------|     x*|-3 + ------|*log(1 - 2*x)|
  |     /     2\                              |          2|       |          2|             |
  |4*log\1 + x /           6*x                \     1 + x /       \     1 + x /             |
4*|------------- - -------------------- - ------------------- + ----------------------------|
  |           3    /     2\           2   /     2\                               2          |
  | (-1 + 2*x)     \1 + x /*(-1 + 2*x)    \1 + x /*(-1 + 2*x)            /     2\           |
  \                                                                      \1 + x /           /

$$4 \left(\frac{x \left(\frac{4 x^{2}}{x^{2} + 1} - 3\right) \log{\left(- 2 x + 1 \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{3 \cdot \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{\left(2 x - 1\right) \left(x^{2} + 1\right)} - \frac{6 x}{\left(2 x - 1\right)^{2} \left(x^{2} + 1\right)} + \frac{4 \log{\left(x^{2} + 1 \right)}}{\left(2 x - 1\right)^{3}}\right)$$

Simplify

The graph

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

The graph:

Piecewise:

The solution