Let $X,Y$ be Banach spaces and $f: X \to Y$. We say that $f$ is Gâteaux differentiable at $x \in X$ if for every $0\neq e \in X$ there exists a a bounded linear operator $l_{x}\in B(X,Y)$ such that \begin{eqnarray} \lim_{t\to 0} \frac{f(x+te)-f(x)-tl_{x}(e)}{t} = 0. \tag{1}\label{1} \end{eqnarray} If $f$ is Gateaux differentiable at $x$, we write \begin{eqnarray} \frac{d}{dt}f(x+te)\bigg{|}_{t=0} \equiv \lim_{t\to 0}\frac{f(x+te)-f(x)}{t} := l_{x}(e) \tag{2}\label{2} \end{eqnarray}
Now, I'd like to understant what does it mean to write \begin{eqnarray} \frac{d^{n}}{dt^{n}}f(x+te) \tag{3},\label{3} \end{eqnarray} i.e. what is exactly the $n$-order Gateaux derivative of $f$. Let's take $n=2$ for simplicity. At first I thought $\frac{d^{2}}{dt^{2}}f(x+te)$ was the derivative of the map $l_{x}: X \to Y$ but, if it is the case, then it should be itself since $l_{x}$ is a linear map, so every higher order Gateaux derivative would be equal to $l_{x}$. So, I think I misunderstood it but I'm really confused to proceed. Can anyone help me, please?