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Similar expressions
1/(1+exp(a*x))
1/(1-exp(-a*x))

The derivative of the function/ 1/(1+exp(-a*x))

Derivative of 1/(1+exp(-a*x))

The solution

You have entered [src]

    1    
---------
     -a*x
1 + e

$$\frac{1}{e^{- a x} + 1}$$

1/(1 + exp((-a)*x))

Detail solution

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

$\frac{a e^{- a x}}{\left(e^{- a x} + 1\right)^{2}}$
Now simplify:

$\frac{a}{4 \cosh^{2}{\left(\frac{a x}{2} \right)}}$

The answer is:

$\frac{a}{4 \cosh^{2}{\left(\frac{a x}{2} \right)}}$

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /        -a*x        -2*a*x  \      
 3 |     6*e         6*e        |  -a*x
a *|1 - --------- + ------------|*e    
   |         -a*x              2|      
   |    1 + e       /     -a*x\ |      
   \                \1 + e    / /      
---------------------------------------
                         2             
              /     -a*x\              
              \1 + e    /

$$\frac{a^{3} \left(1 - \frac{6 e^{- a x}}{1 + e^{- a x}} + \frac{6 e^{- 2 a x}}{\left(1 + e^{- a x}\right)^{2}}\right) e^{- a x}}{\left(1 + e^{- a x}\right)^{2}}$$

Simplify

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Derivative of 1/(1+exp(-a*x))

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