Let $X,Y$ be Banach spaces, define by $F:X\times Y \rightarrow \mathbb{R}$ be a functional, $F_u,F_v$ be Frechet derivative of $F$ with respect to $u$ and $v$ variables. We show that $F$ is Frechet differentiable.
We have
\begin{align} P&=\Vert F\left(u+h,v+l\right)-F\left(u,v\right)-F_u\left(u,v\right)h-F_v\left(u,v\right)l \Vert \\ &= \Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h+F\left(u,v+l\right)-F\left(u,v\right)-F_v\left(u,v\right)l \Vert \\ &\leq \Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h \Vert +\Vert F\left(u,v+l\right)-F\left(u,v\right)-F_v\left(u,v\right)l \Vert \\ &\leq \Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h \Vert +\epsilon\Vert l\Vert \end{align} We need to show that $$\Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v\right)h \Vert \leq \epsilon' \Vert h\Vert \quad ???$$
It follows that $F$ admits $F'\left(u,v\right)=F_u\left(u,v\right)h+F_v\left(u,v\right)l$ is the Frechet derivative. However, we only have that
> $$\Vert F\left(u+h,v+l\right)-F\left(u,v+l\right)-F_u\left(u,v+l\right)h \Vert \leq \epsilon'' \Vert h\Vert \quad ???$$
$F$ is functional which means $F$ is linear, therefor in argument (5th line) rewrite
$$F(u+h,v+l) = F(u+h,v) +F(0,l) $$ and $$ F(u,v+l) = F(u,v) +F(0,l) $$
then cancel out like terms and then considering $F$ is differentiable in first argument, you are done !
For general functions it is easy to provide an example in which $F: R \times R \rightarrow R$ has partial derivatives but not differentiable.