2017-08-22

sigmod求导过程

公式：

f(z) = 1 / (1 + exp(-z))

结果

f(z)' = f(z)(1 − f(z))

推导过程

f(z) = 1 / (1 + exp(-z)) ->     exp(-z) = (1-f(z))/f(z)

f(z)' = (1 / (1 + exp(-z)))'
      = ((1 + exp(-z))^-1)'
      = (-1)*((1 + exp(-z))^-2)*(1 + exp(-z))'
      = -((1 + exp(-z))^-2)*exp(-z)'
      = -((1 + exp(-z))^-2)*(exp(z)^-1)'
      = -((1 + exp(-z))^-2)*(-1)(exp(z)^-2)exp(z)
      = (1 + exp(-z))^-2)*(exp(z)^-1)
      = (f(z)^2)*exp(-z)
      = (f(z)^2)*(1-f(z))/f(z)
      = f(z)*(1-f(z))