Definition:

\[ \frac{dF[f+\epsilon \eta ]}{d \epsilon}\Bigg\rvert_{\epsilon = 0} := \int \frac{\delta F[f]}{\delta f(x')} \eta(x') dx' \]

Alternative definition is:

\[ \frac{\delta F[f(x)]}{\delta f(x')} = \lim_{\epsilon \to 0} \frac{F[f(x) + \epsilon \delta(x-x')] - F[f(x)]}{\epsilon} \]