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

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

The solution

You have entered [src]

           1   
log(x) - ------
              2
         1 + x

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

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

Detail solution

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                         2  
  1        2          8*x   
- -- + --------- - ---------
   2           2           3
  x    /     2\    /     2\ 
       \1 + x /    \1 + x /

$$- \frac{8 x^{2}}{\left(x^{2} + 1\right)^{3}} + \frac{2}{\left(x^{2} + 1\right)^{2}} - \frac{1}{x^{2}}$$

Simplify

The third derivative [src]

  /                       3  \
  |1       12*x       24*x   |
2*|-- - --------- + ---------|
  | 3           3           4|
  |x    /     2\    /     2\ |
  \     \1 + x /    \1 + x / /

$$2 \left(\frac{24 x^{3}}{\left(x^{2} + 1\right)^{4}} - \frac{12 x}{\left(x^{2} + 1\right)^{3}} + \frac{1}{x^{3}}\right)$$

Simplify

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

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Piecewise:

The solution