Taylor series expansion for calculus of variations

Question

$\partial f / \partial u'$ is a functional partial derivative of $f$ with respect to $u'$, where $f = \{ u(x), u'(x) \}$, is it possible to expand the same as a Taylor series? say at point $x_1$ say where $x_2 = x_1 + \delta x$, where $\delta x$ is a small incremental increase from $x_1$