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Similar expressions
y=x^2*ln(1+x^2)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      3                   
   2*x            /     2\
- ------ + 2*x*log\1 - x /
       2                  
  1 - x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

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Piecewise:

The solution