In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})$ $L^p-$smooth if there exists a sequence $F_n$ of continuous, bounded, with continuous and bounded Fréchet derivatives such that

a) $F_n\rightarrow F$ in $L^p(\mu)$, where $\mu$ is the Wiener measure and 

b) $D_WF_n(H)\rightarrow D_WF(H)$ in $L^p(\mu)$, where $D_WF(H)$ denotes the Fréchet derivative of $F$ in direction $H=\int_0^\cdot h_s ds$ for suitable $h$, i.e., $\mathbb{E}[\int_0^1 h_s^2 ds]<\infty$.

In Theorem (2.3) on page 196 he shows that if $X$ is the solution to the SDE 
$dX_t=\sigma(X_t)dW_t$ ($\sigma$ is $mxd$ matrix, $X_0=x_0\in\mathbb{R}^d$, $b,\sigma$ satisfy a linear growth condition and are smooth), then $F(W):=f(X_1)$ ($f$ smooth) is $L^p$-smooth. 

In the proof he uses the following iteration $X_{i+1}(t)=x_0+\int_0^t\sigma(X_i(s))ds$ and then says $DX_{i+1}(t)=I+\int_0^t\sigma'(X_i(s))DX_i(s)dW_s$. He then goes on to show that $X$ is $L^p$ smooth.  

*The idea for the iteration seems to be the following: Generalise the idea of $L^p$ functionals to include Fréchet differentiable $F:C[0,1]\rightarrow \mathbb{R}^d$. Then for suitable $\mathbb{R}^d$-valued time indexed functionals $Y_t$ one can see by approximation that the map  $\int_0^t (\sigma\circ Y_s)dW_s$ will again be an $\mathbb{R}^d$ valued functional on $C[0,1]$.* 

*By assumption $X_0(t)=x_0\in\mathbb{R}^d$ and one easily sees that for $i=0$ $X_1(t)$ is $L^p$ smooth. Now, if this holds for $i=n-1$, it is easily seen to hold for $i=n$ by the just mentioned argument. Hence, each $X_i(t)$ is $L^p$ smooth. $X_i(t)$ converges to $X(t)$ in $L^p$.*

Now, how can I conclude from this that also the Fréchet derivatives of $X_i(t)$ converge to the Fréchet derivative of $X(t)$? I suppose that this is what I need the convergence of the above $DX_i(t)$ to $DX(t)$ for, but I do not understand how this helps.