Computing Frechet derivative Let $F:C^{([-1,1])}\rightarrow C^{([-1,1])}$ defined as $F(q)(x)=q^{2}(-x)$.
I want to find the derivative of this function my current stratergy is to use the directional derivative so $dF(q)(x)h= \frac{d}{d\epsilon}|_{\epsilon=0} (q+\epsilon h)^{2}(-x)=2hq(-x)$. 
Is this correct of do i need to do something different?

Let $\mathrm{H} = \mathscr{C}(\mathrm{I}),$ the space of continuous real-valued functions defined on the compact interval $\mathrm{I}$ the with norm of suprema. Let $s$ be the function $\mathrm{H} \to \mathrm{H}$ given by $s(f)(x) = f(-x),$ $z$ the function $\mathrm{H} \to \mathrm{H} \times \mathrm{H}$ given by $q \mapsto (q, q)$ and let $B$ the function $\mathrm{H} \times \mathrm{H} \to \mathrm{H}$ given by $B(f,g) = fg.$ It is easy to see $s$ and $z$ are continuous linear functions and $B$ is a continuous bilinear (with $\|B\| \leq 1$). The function you want to differentiat is given $F = B \circ  z \circ s.$ By the chain rule, $F'(q) = B'(s(q), s(q)) \circ z \circ s;$ hence, $F'(q) \cdot h = B'(s(q), s(q)) \cdot (s(h), s(h)) = 2 s(q) s(h),$ if you evaluate in a point $x$ this becomes $\big(F'(q) \cdot h \big)(x) = 2q(-x)h(-x).$ Q.E.D.
