Chain rule with exponential map

Question

Let $G$ be a Lie group, $X,Y \in $ Lie($G$) and $f \in C^{\infty}(G)$. Then, by the chain rule:

$\frac{d}{dt} \Bigr \lvert_{t=0} f(exp(tX)exp(tY)) = \frac{d}{dt} \Bigr \lvert_{t=0} f(exp(tX)exp(0 \cdot Y))+ \frac{d}{dt} \Bigr \lvert_{t=0} f(exp(0 \cdot X)exp(tY)) = \frac{d}{dt} \Bigr \lvert_{t=0} f(exp(tX)) + \frac{d}{dt} \Bigr \lvert_{t=0} f(exp(tY)) = Xf + Yf$.

I don't get the first equality. Could someone give me a hand with this please?

Answer 1

Introduce the function $F(s,t) = f\big(\exp(sX)\exp(tY)\big)$ and differentiate $F(s,t)$. Then set $\phi(t) = (t,t)$ and apply the chain rule to differentiate $F(\phi(t))$.