The following exercise is from Naive Lie Theory by Stillwell, and is designed (I assume) to illustrate how non-commutativity of quaternions affects the product rule.
The definition of derivative for any function $c(t)$ of a real variable $t$ is $$c'(t) = \lim_{\Delta t \to 0} \frac{c(t + \Delta t) - c(t)}{\Delta t}.$$
(1) By imitating the usual proof of the product rule, show that if $c(t) = a(t)b(t)$ then $$c'(t) = a'(t)b(t) + a(t)b'(t).$$ (Do not assume the product rule is commutative.)
(2) Show also that if $c(t) = a(t)^{-1}$, and $a(0) = 1$, then $c'(0)= -a'(0),$ again without assuming that the product is commutative.
(3) Show, however, that if $c(t)=a(t)^2$ then $c'(t)$ is not equal to $2a(t)a'(t)$ for a certain quaternionic-valued function $a(t)$.
Parts (1) and (2) are straightforward, but I just cannot get the certain quaternionic-valued function for part (3). Does anyone know what it is? For the function $a(t)$ I've tried: $it, jt, (i+j)t, ijt, e^jt$ and a whole bunch of others without success. If anyone knows it, i'd be grateful, thanks.
If $a(t)=it+j$, then $c(t)=a^2(t)=-1-t^2$, and therefore$$2a(t)a'(t)=2(it+j)i=-2t-k,$$which is not $c'(t)$.