I have been reading 'Lagrangians and Hamiltonians' by Patrick Hamill and it states that the functional derivative of a function $\phi = \phi(y, y', x)$ w.r.t. $y$, where $y=y(x)$ and $y' = \frac{dy}{dx}$ is given by
$$ \frac{\delta \phi}{\delta y} = \frac{\partial \phi}{\partial y} - \frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial \phi}{\partial y'}. $$
However if I use the definition of the functional derivative
$$ \frac{\delta F[y(x)]}{\delta y(x')} = \lim_{\varepsilon \rightarrow 0} \frac{1}{\varepsilon}(F[y + \varepsilon \delta(x-x')] - F[y]),$$
I get
$$ \frac{\delta \phi(y,y',x)}{\delta y(x')} = \bigg(\frac{\partial \phi}{\partial y} - \frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial \phi}{\partial y'}\Bigg) \delta(x'-x). $$
Why are they different?
Thanks