Can you help me understanding how author got to equation 1.12, and what is phi(X)function. (https://i.stack.imgur.com/16LOQ.jpg)
$$J[f] = \int [f(y)]^p \phi{(y)} d{y}$$
$$\frac{\delta{J[f]}}{\delta{f(x)}} = \lim_{\epsilon\to0}\frac{[\int [f(y) + \epsilon\delta{(y-x)}]^p \phi{(y)}d{y} - \int[f(y)]^p \phi{(y)}dy ]}{\epsilon} $$ $$\frac{\delta{J[f]}}{\delta{f(x)}} = p[f(x)]^{p-1} \phi(x) \space\space\space (1.12)$$
Use the generalized binomial theorem:\begin{align} \frac{\delta{J[f]}}{\delta{f(x)}} &= \lim_{\epsilon\to0}\frac1\varepsilon\left(\int [f(y) + \epsilon\delta{(y-x)}]^p \phi{(y)}\,dy - \int[f(y)]^p \phi{(y)}\,dy\right)\\ &= \lim_{\epsilon\to0}\frac1\varepsilon\left(\int \sum_{k=0}^\infty{p \choose k} [\varepsilon \delta(y-x)]^k [f(y)]^{p-k} \phi{(y)}\,dy - \int[f(y)]^p \phi{(y)}\,dy\right)\\ &= \lim_{\epsilon\to0}\left[p\int \delta(y-x)[f(y)]^{p-1}\phi(y)\,dy + \sum_{k=2}^\infty{p \choose k} \varepsilon^{k-1} \int [\delta(y-x)]^k [f(y)]^{p-k} \phi{(y)}\,dy\right]\\ &= p\int \delta(y-x)[f(y)]^{p-1}\phi(y)\,dy\\ &= p[f(x)]^{p-1}\phi(x) \end{align}
$\phi$ is just an arbitrary function used to define the functional $J$.