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

Derivative of y=ln(x-1/x+1)

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

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from to

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The solution

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

  1. Let u=(x−1x)+1u = \left(x - \frac{1}{x}\right) + 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−1x)+1)\frac{d}{d x} \left(\left(x - \frac{1}{x}\right) + 1\right):

    1. Differentiate (x−1x)+1\left(x - \frac{1}{x}\right) + 1 term by term:

      1. Differentiate x−1xx - \frac{1}{x} term by term:

        1. Apply the power rule: xx goes to 11

        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: 1x2\frac{1}{x^{2}}

        The result is: 1+1x21 + \frac{1}{x^{2}}

      2. The derivative of the constant 11 is zero.

      The result is: 1+1x21 + \frac{1}{x^{2}}

    The result of the chain rule is:

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

  4. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-2525
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      1  
  1 + -- 
       2 
      x  
---------
    1    
x - - + 1
    x
1+1x2(x−1x)+1\frac{1 + \frac{1}{x^{2}}}{\left(x - \frac{1}{x}\right) + 1}
Simplify
The second derivative [src] 
 /             2\ 
 |     /    1 \ | 
 |     |1 + --| | 
 |     |     2| | 
 |2    \    x / | 
-|-- + ---------| 
 | 3           1| 
 |x    1 + x - -| 
 \             x/ 
------------------
            1     
    1 + x - -     
            x
−(1+1x2)2x+1−1x+2x3x+1−1x- \frac{\frac{\left(1 + \frac{1}{x^{2}}\right)^{2}}{x + 1 - \frac{1}{x}} + \frac{2}{x^{3}}}{x + 1 - \frac{1}{x}}
Simplify
The third derivative [src] 
  /              3                   \
  |      /    1 \          /    1 \  |
  |      |1 + --|        3*|1 + --|  |
  |      |     2|          |     2|  |
  |3     \    x /          \    x /  |
2*|-- + ------------ + --------------|
  | 4              2    3 /        1\|
  |x    /        1\    x *|1 + x - -||
  |     |1 + x - -|       \        x/|
  \     \        x/                  /
--------------------------------------
                      1               
              1 + x - -               
                      x
2((1+1x2)3(x+1−1x)2+3(1+1x2)x3(x+1−1x)+3x4)x+1−1x\frac{2 \left(\frac{\left(1 + \frac{1}{x^{2}}\right)^{3}}{\left(x + 1 - \frac{1}{x}\right)^{2}} + \frac{3 \left(1 + \frac{1}{x^{2}}\right)}{x^{3} \left(x + 1 - \frac{1}{x}\right)} + \frac{3}{x^{4}}\right)}{x + 1 - \frac{1}{x}}
Simplify
The graph
Derivative of y=ln(x-1/x+1)
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