Derivative of ln(x+1/x)

You have entered [src]

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

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

log(x + 1/x)

Detail solution

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

$\frac{1 - \frac{1}{x^{2}}}{x + \frac{1}{x}}$
Now simplify:

$\frac{x^{2} - 1}{x^{3} + x}$

The answer is:

$\frac{x^{2} - 1}{x^{3} + x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\frac{1 - \frac{1}{x^{2}}}{x + \frac{1}{x}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /               3             \
  |       /    1 \      /    1 \|
  |       |1 - --|    3*|1 - --||
  |       |     2|      |     2||
  |  3    \    x /      \    x /|
2*|- -- + --------- - ----------|
  |   4           2    3 /    1\|
  |  x     /    1\    x *|x + -||
  |        |x + -|       \    x/|
  \        \    x/              /
---------------------------------
                  1              
              x + -              
                  x

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ln(x+1/x)

The graph:

Piecewise:

The solution