Let f: ℝ → ℝ be the function f(x) = 1/(1 + e⁻ˣ). The value of the derivative of f at x where f(x) = 0.4 is…

Question

Let f: ℝ → ℝ be the function f(x) = 1/(1 + e⁻ˣ).

The value of the derivative of f at x where f(x) = 0.4 is ______ (rounded off to two decimal places).

Note: ℝ denotes the set of real numbers.

Accepted Answer

Correct answer: 0.24 — Solution: use the standard derivative formula for the logistic (sigmoid) function. Derivative formula: f'(x) = f(x) (1 - f(x)). Given f(x) = 0.4, substitute to get f'(x) = 0.4 × (1 - 0.4) = 0.4 × 0.6 = 0.24. Rounded to two decimal places: 0.24.

Let f: ℝ → ℝ be the function f(x) = 1/(1 + e⁻ˣ). The value of the derivative…