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Derivative of:
Derivative of cos(x/3)
Derivative of √x
Derivative of xcos(x)
Derivative of 50/x
Graphing y =:
ln(1/(x^2+1))
Identical expressions
ln(one /(x^ two + one))
ln(1 divide by (x squared plus 1))
ln(one divide by (x to the power of two plus one))
ln(1/(x2+1))
ln1/x2+1
ln(1/(x²+1))
ln(1/(x to the power of 2+1))
ln1/x^2+1
ln(1 divide by (x^2+1))
Similar expressions
ln(1/(x^2-1))

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

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

The solution

You have entered [src]

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

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

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

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify