Originally Posted by

**zzzoak** May be this helps. The definition is

$\displaystyle

\displaystyle

d_h f=lim_{\epsilon \to 0} \frac{f(x+\epsilon h)-f(x)}{\epsilon}

$

where

$\displaystyle

x \; and \; h \ \in \ D.

$

So we have

$\displaystyle

\displaystyle

d_h f=lim_{\epsilon \to 0} \frac{f(x+\epsilon h)(t)-f(x)(t)}{\epsilon}=

$

$\displaystyle

\displaystyle

=lim_{\epsilon \to 0} \frac{1}{\epsilon}\; ( \; \int_0^1 g(t,x(s)+\epsilon h)ds-\int_0^1 g(t,x(s))ds \; )=\; \int_0^1 \frac{\partial g(t,x(s))}{\partial x} \ h \ ds \ .

$