  Let $X_1, \dots, X_n$ be Banach spaces and $f$ a real function on $X=X_1
\times\dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which
$f$ is partially Fr\' echet differentiable but is not Fr\' echet
differentiable. Our results imply that if $X_1, \dots, X_{n-1}$ are Asplund
spaces and $f$ is continuous (resp. Lipschitz) on $X$, then $A_f$ is a first
category set (resp. a $\sigma$-upper porous set). We also prove that if $X$,
$Y$ are separable Banach spaces and $f:X \to Y$ is a Lipschitz mapping, then
the set of all points $x \in X$ at which $f$ is G\^ ateaux differentiable, is
Fr\' echet differentiable along a closed subspace of finite codimension but is
not Fr\' echet differentiable, is $\sigma$-upper porous. A number of related
more general results are also proved.
