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ln(1-x^2)

Derivative of ln(1-x^2)

Function f() - derivative -N order at the point
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from to

Piecewise:

The solution

You have entered [src]
   /     2\
log\1 - x /
log(1x2)\log{\left(1 - x^{2} \right)}
log(1 - x^2)
Detail solution
  1. Let u=1x2u = 1 - x^{2}.

  2. The derivative of log(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddx(1x2)\frac{d}{d x} \left(1 - x^{2}\right):

    1. Differentiate 1x21 - x^{2} term by term:

      1. The derivative of the constant 11 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: x2x^{2} goes to 2x2 x

        So, the result is: 2x- 2 x

      The result is: 2x- 2 x

    The result of the chain rule is:

    2x1x2- \frac{2 x}{1 - x^{2}}

  4. Now simplify:

    2xx21\frac{2 x}{x^{2} - 1}


The answer is:

2xx21\frac{2 x}{x^{2} - 1}

The graph
02468-8-6-4-2-1010-2525
The first derivative [src]
 -2*x 
------
     2
1 - x 
2x1x2- \frac{2 x}{1 - x^{2}}
The second derivative [src]
  /         2 \
  |      2*x  |
2*|1 - -------|
  |          2|
  \    -1 + x /
---------------
          2    
    -1 + x     
2(2x2x21+1)x21\frac{2 \left(- \frac{2 x^{2}}{x^{2} - 1} + 1\right)}{x^{2} - 1}
The third derivative [src]
    /          2 \
    |       4*x  |
4*x*|-3 + -------|
    |           2|
    \     -1 + x /
------------------
             2    
    /      2\     
    \-1 + x /     
4x(4x2x213)(x21)2\frac{4 x \left(\frac{4 x^{2}}{x^{2} - 1} - 3\right)}{\left(x^{2} - 1\right)^{2}}
The graph
Derivative of ln(1-x^2)