Derivative of the product of operators

Question

I'm asked to show that

$\frac{d(\hat{A}\hat{B})}{d\lambda} = \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$

With $\lambda$ a continuous parameter

Should I use the definition

$\frac{d\hat{A}}{d\lambda} = \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon) - \hat{A}(\lambda)}{\epsilon}$

applied to $\hat{A}\hat{B}$ like

$\frac{d(\hat{A}\hat{B})}{d\lambda} = \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon)\hat{B}(\lambda + \epsilon) - \hat{A}(\lambda)\hat{B}(\lambda)}{\epsilon}$

and do some algebra to get the RHS of the first equation, or I'm missing something?

Thanks for your time

Another interesting derivative to pay attention to is: $\frac{d}{d\lambda}\exp(\hat{A}(\lambda) )$?

Answer 1

$$A(\lambda+\epsilon)B(\lambda+\epsilon) = (A(\lambda) + \epsilon \dot{A} )(B(\lambda) +\epsilon \dot B ) = A(\lambda)B(\lambda) + \epsilon(\dot AB+A\dot B) + o(\epsilon^2)$$

Answer 2

Here we will only consider the added last subquestion (v4):

$$\tag{1} \frac{d}{d\lambda}e^{\hat{A}} ~=~ \int_0^1\!ds~e^{(1-s)\hat{A}}\frac{d\hat{A}}{d\lambda}e^{s\hat{A}} .$$

The identity (1) follows by setting $t=1$ in the following identity

$$\tag{2} e^{-t\hat{A}} \frac{d}{d\lambda}e^{t\hat{A}} ~=~ \int_0^t\!ds~e^{-s\hat{A}}\frac{d\hat{A}}{d\lambda}e^{s\hat{A}} .$$

To prove equation (2), first note that (2) is trivially true for $t=0$. Secondly, note that a differentiation wrt. $t$ on both sides of (2) produces the same expression

$$\tag{3}e^{-t\hat{A}}\frac{d\hat{A}}{d\lambda}e^{t\hat{A}},$$

where we use the fact that

$$\tag{4}\frac{d}{dt}e^{t\hat{A}}~=~\hat{A}e^{t\hat{A}}~=~e^{t\hat{A}}\hat{A}.$$

So the two sides of eq.(2) must be equal.