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

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   /    1    \
log|x - - + 1|
   \    x    /

$$\log{\left(\left(x - \frac{1}{x}\right) + 1 \right)}$$

log(x - 1/x + 1)

Detail solution

Let $u = \left(x - \frac{1}{x}\right) + 1$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x - \frac{1}{x}\right) + 1\right)$ :
1. Differentiate $\left(x - \frac{1}{x}\right) + 1$ term by term:
  1. Differentiate $x - \frac{1}{x}$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
      So, the result is: $\frac{1}{x^{2}}$
    The result is: $1 + \frac{1}{x^{2}}$
  2. The derivative of the constant $1$ is zero.
  The result is: $1 + \frac{1}{x^{2}}$
The result of the chain rule is:

$\frac{1 + \frac{1}{x^{2}}}{\left(x - \frac{1}{x}\right) + 1}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1  
  1 + -- 
       2 
      x  
---------
    1    
x - - + 1
    x

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

Simplify

The second derivative [src]

 /             2\ 
 |     /    1 \ | 
 |     |1 + --| | 
 |     |     2| | 
 |2    \    x / | 
-|-- + ---------| 
 | 3           1| 
 |x    1 + x - -| 
 \             x/ 
------------------
            1     
    1 + x - -     
            x

$$- \frac{\frac{\left(1 + \frac{1}{x^{2}}\right)^{2}}{x + 1 - \frac{1}{x}} + \frac{2}{x^{3}}}{x + 1 - \frac{1}{x}}$$

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

$$\frac{2 \left(\frac{\left(1 + \frac{1}{x^{2}}\right)^{3}}{\left(x + 1 - \frac{1}{x}\right)^{2}} + \frac{3 \left(1 + \frac{1}{x^{2}}\right)}{x^{3} \left(x + 1 - \frac{1}{x}\right)} + \frac{3}{x^{4}}\right)}{x + 1 - \frac{1}{x}}$$

Simplify

The graph

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

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The solution