For the following activation functions, show that (a) if (v) = 1/(1 + eav), then (v) =

For the following activation functions, show that

(a) if φ(v) = 1/(1 + e−av), then φ′(v) = aφ(v)(1 − φ(v));

(b) if φ(v) = a tanh(bv), then φ′(v) = b a (a2 − φ2(v));

(c) if φ(v) = 1 2a log cosh(a(v+1))

cosh(a(v−1)) , then

φ′(v) = (tanh(a(v + 1)) − tanh(a(v − 1)))/2.