Derivative of ln(1+x²)

The solution

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   /     2\
log\1 + x /

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

log(1 + x^2)

Detail solution

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

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

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
1 + x

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

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

The graph

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Derivative of ln(1+x²)

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The solution