Compute the derivative f′(x)𝑓′(𝑥) of the logistic sigmoid11+e−x11+𝑒−𝑥111e−x1𝑒−𝑥0

The derivative of the logistic sigmoid function f(x) = 1 / (1 + e^-x) can be computed as follows:

Step 1: Write down the function f(x) = 1 / (1 + e^-x)

Step 2: Recognize that this is a composition of functions. The outer function is f(u) = 1/u and the inner function is g(x) = 1 + e^-x.

Step 3: Compute the derivative of the outer function evaluated at the inner function. The derivative of f(u) = 1/u is f'(u) = -1/u^2. Evaluating this at u = g(x) gives f'(g(x)) = -1/(1 + e^-x)^2.

Step 4: Compute the derivative of the inner function. The derivative of g(x) = 1 + e^-x is g'(x) = -e^-x.

Step 5: Apply the chain rule. The chain rule states that the derivative of a composition of functions is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This gives f'(x) = f'(g(x)) * g'(x) = -1/(1 + e^-x)^2 * -e^-x.

Step 6: Simplify the expression. The negatives cancel and the e^-x in the numerator and denominator cancel to give f'(x) = e^-x / (1 + e^-x)^2.

Step 7: Recognize that e^-x / (1 + e^-x) is just the original function f(x) and that (1 + e^-x)^2 is the original function squared. This gives f'(x) = f(x) * (1 - f(x)).

So, the derivative of the logistic sigmoid function is f'(x) = f(x) * (1 - f(x)).