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 /
$$\log{\left(1 - e^{x} \right)}$$
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 a constant times a function is the constant times the derivative of the function.

        1. The derivative of is itself.

        So, the result is:

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The first derivative [src]
   x  
 -e   
------
     x
1 - E 
$$- \frac{e^{x}}{1 - e^{x}}$$
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)