How do you find the first and second derivative of y=1/(1+e^(-x))?

Guillermo May

Guillermo May

Answered question

2023-03-13

How to find the first and second derivative of y = 1 1 + e - x ?

Answer & Explanation

wmbishopxxviieg7

wmbishopxxviieg7

Beginner2023-03-14Added 5 answers

d y d x = e - x ( 1 + e - x ) 2 , d 2 y d x 2 = e - 2 x - e - x ( 1 + e - x ) 3
Solution:
Reminder | 2 2 d d x ( e - x ) = - e - x 2 2 | ̲ ¯
There are 2 approaches to differentiating this function.
(1) Using the quotient rule
( 2 ) expressing y = ( 1 + e - x ) - 1 and use chain rule
I'll use approach (1), but you could try approach (2). The outcome will be the same.
differentiate using the quotient rule
Given y = g ( x ) h ( x ) then
| 2 2 d y d x = h ( x ) g ( x ) - g ( x ) h ( x ) ( h ( x ) ) 2 2 2 | ̲ ¯
g ( x ) = 1 g ( x ) = 0
h ( x ) = 1 + e - x h ( x ) = - e - x
d y d x = ( 1 + e - x ) .0 - 1 . ( - e - x ) ( 1 + e - x ) 2 = e - x ( 1 + e - x ) 2
To find d 2 y d x 2 differentiate d y d x
differentiate using the quotient rule/chain rule
here g ( x ) = e - x g ( x ) = - e - x
h ( x ) = ( 1 + e - x ) 2 h ( x ) = 2 ( 1 + e - x ) . ( - e - x )
h ( x ) = - 2 e - x ( 1 + e - x )
d 2 y d x 2 = ( 1 + e - x ) 2 ( - e - x ) - ( e - x ) . ( - 2 e - x ( 1 + e - x ) ) ( 1 + e - x ) 4
= - e - x ( 1 + e - x ) 2 + 2 e - 2 x ( 1 + e - x ) ( 1 + e - x ) 4
= e - x ( 1 + e - x ) ( 2 e - x - 1 - e - x ) ( 1 + e - x ) 4 factoring
= e - x ( 1 + e - x ) ( e - x - 1 ) ( 1 + e - x ) 3
= e - 2 x - e - x ( 1 + e - x ) 3

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