Mister Exam

Derivative of ln(1+e^x)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /     x\
log\1 + E /
$$\log{\left(e^{x} + 1 \right)}$$
log(1 + E^x)
Detail solution
  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of is itself.

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
   x  
  e   
------
     x
1 + E 
$$\frac{e^{x}}{e^{x} + 1}$$
The second derivative [src]
/       x  \   
|      e   |  x
|1 - ------|*e 
|         x|   
\    1 + e /   
---------------
          x    
     1 + e     
$$\frac{\left(1 - \frac{e^{x}}{e^{x} + 1}\right) e^{x}}{e^{x} + 1}$$
The third derivative [src]
/        x         2*x \   
|     3*e       2*e    |  x
|1 - ------ + ---------|*e 
|         x           2|   
|    1 + e    /     x\ |   
\             \1 + e / /   
---------------------------
                x          
           1 + e           
$$\frac{\left(1 - \frac{3 e^{x}}{e^{x} + 1} + \frac{2 e^{2 x}}{\left(e^{x} + 1\right)^{2}}\right) e^{x}}{e^{x} + 1}$$
The graph
Derivative of ln(1+e^x)