Computation of a derivative in a Banach algebra

Question

Let A be a complex unital Banach algebra with the unit e and φ : A → A a continuous antilinear mapping such that φ(exp(x)) is invertible, for every x in A. Let f be the following map: f(λ) = [φ(exp(λx))]y[φ(exp(λx))]−1

1. How can I compute the derivative of f by λ, where λ lies is a complex number. I don't know how to compute the derivative of [φ(exp(λx))]−1.
2. How can I show that f is harmonic?

Many thanks in advance for your help!