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[tex]f: \mathbb{R}^{n} \rightarrow \mathbb{R}[/tex]

I know that the first derivative of this function is a linear map [tex]\lambda: \mathbb{R}^{n}\rightarrow\mathbb{R}[/tex]. However, when we take the second-derivative of [tex]\lambda[/tex], some questions arise for me:

1.) If we are taking this derivative when considering [tex]\lambda[/tex] as a linear function, then we'd just get back [tex]\lambda[/tex], which isn't the case. So how are we interpreting the first derivative when taking a second?

2.) In general, why do we say that [tex]D^{k}f:\mathbb{R}^{n^{k}}\rightarrow\mathbb{R}[/tex] and not [tex]D^{k}f:\mathbb{R}^{n}\rightarrow\mathbb{R}[/tex] ??

Thanks in advance.