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The derivative of the function/ 1/4(ln((x-1)/(x+1)))

Derivative of 1/4(ln((x-1)/(x+1)))

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
   /x - 1\
log|-----|
   \x + 1/
----------
    4
log⁡(x−1x+1)4\frac{\log{\left(\frac{x - 1}{x + 1} \right)}}{4} 
log((x - 1)/(x + 1))/4
Detail solution

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

    1. Let u=x−1x+1u = \frac{x - 1}{x + 1}.

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

    3. Then, apply the chain rule. Multiply by ddxx−1x+1\frac{d}{d x} \frac{x - 1}{x + 1}:

      1. Apply the quotient rule, which is:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

        f(x)=x−1f{\left(x \right)} = x - 1 and g(x)=x+1g{\left(x \right)} = x + 1.

        To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Differentiate x−1x - 1 term by term:

          1. The derivative of the constant −1-1 is zero.

          2. Apply the power rule: xx goes to 11

          The result is: 11

        To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Differentiate x+1x + 1 term by term:

          1. The derivative of the constant 11 is zero.

          2. Apply the power rule: xx goes to 11

          The result is: 11

        Now plug in to the quotient rule:

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

      The result of the chain rule is:

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

    So, the result is: x+12(x−1)(x+1)2\frac{x + 1}{2 \left(x - 1\right) \left(x + 1\right)^{2}}

  2. Now simplify:

    12(x2−1)\frac{1}{2 \left(x^{2} - 1\right)}

The answer is:

12(x2−1)\frac{1}{2 \left(x^{2} - 1\right)}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        /  1      x - 1  \
(x + 1)*|----- - --------|
        |x + 1          2|
        \        (x + 1) /
--------------------------
        4*(x - 1)
(x+1)(−x−1(x+1)2+1x+1)4(x−1)\frac{\left(x + 1\right) \left(- \frac{x - 1}{\left(x + 1\right)^{2}} + \frac{1}{x + 1}\right)}{4 \left(x - 1\right)}
Simplify
The second derivative [src] 
/     -1 + x\ /  1       1   \
|-1 + ------|*|----- + ------|
\     1 + x / \1 + x   -1 + x/
------------------------------
          4*(-1 + x)
(x−1x+1−1)(1x+1+1x−1)4(x−1)\frac{\left(\frac{x - 1}{x + 1} - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x - 1}\right)}{4 \left(x - 1\right)}
Simplify
The third derivative [src] 
 /     -1 + x\ /   1           1              1        \ 
-|-1 + ------|*|-------- + --------- + ----------------| 
 \     1 + x / |       2           2   (1 + x)*(-1 + x)| 
               \(1 + x)    (-1 + x)                    / 
---------------------------------------------------------
                        2*(-1 + x)
−(x−1x+1−1)(1(x+1)2+1(x−1)(x+1)+1(x−1)2)2(x−1)- \frac{\left(\frac{x - 1}{x + 1} - 1\right) \left(\frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x - 1\right) \left(x + 1\right)} + \frac{1}{\left(x - 1\right)^{2}}\right)}{2 \left(x - 1\right)}
Simplify
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