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The derivative of the function/ ln(1+x)/(1-x)+2arctg1/x

Derivative of ln(1+x)/(1-x)+2arctg1/x

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

You have entered [src] 
log(1 + x)   2*atan(1)
---------- + ---------
  1 - x          x
log⁡(x+1)1−x+2atan⁡(1)x\frac{\log{\left(x + 1 \right)}}{1 - x} + \frac{2 \operatorname{atan}{\left(1 \right)}}{x} 
log(1 + x)/(1 - x) + (2*atan(1))/x
Detail solution

  1. Differentiate log⁡(x+1)1−x+2atan⁡(1)x\frac{\log{\left(x + 1 \right)}}{1 - x} + \frac{2 \operatorname{atan}{\left(1 \right)}}{x} term by term:

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

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

      1. Let u=x+1u = 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 ddx(x+1)\frac{d}{d x} \left(x + 1\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

        The result of the chain rule is:

        1x+1\frac{1}{x + 1}

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

      1. Differentiate 1−x1 - x 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: xx goes to 11

          So, the result is: −1-1

        The result is: −1-1

      Now plug in to the quotient rule:

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

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

      1. Apply the power rule: 1x\frac{1}{x} goes to −1x2- \frac{1}{x^{2}}

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

    The result is: 1−xx+1+log⁡(x+1)(1−x)2−2atan⁡(1)x2\frac{\frac{1 - x}{x + 1} + \log{\left(x + 1 \right)}}{\left(1 - x\right)^{2}} - \frac{2 \operatorname{atan}{\left(1 \right)}}{x^{2}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-250250
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       1          log(1 + x)   2*atan(1)
--------------- + ---------- - ---------
(1 + x)*(1 - x)           2         2   
                   (1 - x)         x
1(1−x)(x+1)+log⁡(x+1)(1−x)2−2atan⁡(1)x2\frac{1}{\left(1 - x\right) \left(x + 1\right)} + \frac{\log{\left(x + 1 \right)}}{\left(1 - x\right)^{2}} - \frac{2 \operatorname{atan}{\left(1 \right)}}{x^{2}}
Simplify
The second derivative [src] 
        1           2*log(1 + x)           2           4*atan(1)
----------------- - ------------ + ----------------- + ---------
       2                     3                     2        3   
(1 + x) *(-1 + x)    (-1 + x)      (1 + x)*(-1 + x)        x
1(x−1)(x+1)2+2(x−1)2(x+1)−2log⁡(x+1)(x−1)3+4atan⁡(1)x3\frac{1}{\left(x - 1\right) \left(x + 1\right)^{2}} + \frac{2}{\left(x - 1\right)^{2} \left(x + 1\right)} - \frac{2 \log{\left(x + 1 \right)}}{\left(x - 1\right)^{3}} + \frac{4 \operatorname{atan}{\left(1 \right)}}{x^{3}}
Simplify
The third derivative [src] 
  12*atan(1)           6                   3                    2           6*log(1 + x)
- ---------- - ----------------- - ------------------ - ----------------- + ------------
       4                       3          2         2          3                     4  
      x        (1 + x)*(-1 + x)    (1 + x) *(-1 + x)    (1 + x) *(-1 + x)    (-1 + x)
−2(x−1)(x+1)3−3(x−1)2(x+1)2−6(x−1)3(x+1)+6log⁡(x+1)(x−1)4−12atan⁡(1)x4- \frac{2}{\left(x - 1\right) \left(x + 1\right)^{3}} - \frac{3}{\left(x - 1\right)^{2} \left(x + 1\right)^{2}} - \frac{6}{\left(x - 1\right)^{3} \left(x + 1\right)} + \frac{6 \log{\left(x + 1 \right)}}{\left(x - 1\right)^{4}} - \frac{12 \operatorname{atan}{\left(1 \right)}}{x^{4}}
Simplify
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