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Derivative of log(1+1/(x^2))
Graphing y =:
log(1+1/(x^2))
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Similar expressions
log(1-1/(x^2))

The derivative of the function/ log(1+1/(x^2))

Derivative of log(1+1/(x^2))

The solution

You have entered [src]

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

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

log(1 + 1/(x^2))

Detail solution

Let $u = 1 + \frac{1}{x^{2}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 + \frac{1}{x^{2}}\right)$ :
1. Differentiate $1 + \frac{1}{x^{2}}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = x^{2}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{2}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    The result of the chain rule is:
    
    $- \frac{2}{x^{3}}$
  The result is: $- \frac{2}{x^{3}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /          4              9     \
4*|-6 - ------------ + -----------|
  |                2    2 /    1 \|
  |      4 /    1 \    x *|1 + --||
  |     x *|1 + --|       |     2||
  |        |     2|       \    x /|
  \        \    x /               /
-----------------------------------
             5 /    1 \            
            x *|1 + --|            
               |     2|            
               \    x /

$$\frac{4 \left(-6 + \frac{9}{x^{2} \left(1 + \frac{1}{x^{2}}\right)} - \frac{4}{x^{4} \left(1 + \frac{1}{x^{2}}\right)^{2}}\right)}{x^{5} \left(1 + \frac{1}{x^{2}}\right)}$$

Simplify

The graph

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Derivative of log(1+1/(x^2))

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The solution