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The derivative of the function/ log(1-x^2)/log(5)

Derivative of log(1-x^2)/log(5)

Function f() - derivative -N order at the point
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The graph:

from to

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The solution

You have entered [src] 
   /     2\
log\1 - x /
-----------
   log(5)
log⁡(1−x2)log⁡(5)\frac{\log{\left(1 - x^{2} \right)}}{\log{\left(5 \right)}} 
log(1 - x^2)/log(5)
Detail solution

  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Let u=1−x2u = 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(1−x2)\frac{d}{d x} \left(1 - x^{2}\right):

      1. Differentiate 1−x21 - 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:

      −2x1−x2- \frac{2 x}{1 - x^{2}}

    So, the result is: −2x(1−x2)log⁡(5)- \frac{2 x}{\left(1 - x^{2}\right) \log{\left(5 \right)}}

  2. Now simplify:

    2x(x2−1)log⁡(5)\frac{2 x}{\left(x^{2} - 1\right) \log{\left(5 \right)}}

The answer is:

2x(x2−1)log⁡(5)\frac{2 x}{\left(x^{2} - 1\right) \log{\left(5 \right)}}

The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      -2*x     
---------------
/     2\       
\1 - x /*log(5)
−2x(1−x2)log⁡(5)- \frac{2 x}{\left(1 - x^{2}\right) \log{\left(5 \right)}}
Simplify
The second derivative [src] 
   /          2 \
   |       2*x  |
-2*|-1 + -------|
   |           2|
   \     -1 + x /
-----------------
 /      2\       
 \-1 + x /*log(5)
−2(2x2x2−1−1)(x2−1)log⁡(5)- \frac{2 \left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right) \log{\left(5 \right)}}
Simplify
The third derivative [src] 
    /          2 \
    |       4*x  |
4*x*|-3 + -------|
    |           2|
    \     -1 + x /
------------------
         2        
/      2\         
\-1 + x / *log(5)
4x(4x2x2−1−3)(x2−1)2log⁡(5)\frac{4 x \left(\frac{4 x^{2}}{x^{2} - 1} - 3\right)}{\left(x^{2} - 1\right)^{2} \log{\left(5 \right)}}
Simplify
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