The question may not be too well-posed, but loosely speaking, suppose $L:W^{1,p}(\mathbb R)\to L^p(\mathbb R)$ is a (possibly nonlinear) first order differential differential operator, such that all the "coefficients" in $L$ are smooth. Formally, I can compute the directional derivative of $L$ at $f$, along the direction $g\in T_fW^{1,p}(\mathbb R)\cong W^{1,p}(\mathbb R)$, as
$$d_fL(g)=\left .\frac{d}{dt}\right|_{t=0}L(f+tg)$$
Without knowing more information, are the following statement true?
(a). Is $d_fL:T_fW^{1,p}(\mathbb R)\to L^p(\mathbb R)$ a bounded operator?
(b). If (a) is true, is $dL:W^{1,p}\to B(T_fW^{1,p},L^p)$ a bounded operator? In other word, is $L$ a $C^1$-operator (or even $C^{\infty}$) in the Frechet derivative sense?
Motivation: I was reading "Morse theory and Floer Homology" by M.Audin & M.Damian, and they define the Floer operator $\mathcal F$ and compute it's derivative $d\mathcal F$ explicitly (in chapter 8). Since they use Banach implicit function theorem, I assume that we need $\mathcal F$ to be $C^1$ in the Frechet sense. Here $\mathcal F$ is nonlinear elliptic.