begin{aligned}{\frac {\mathrm {d} \Phi }{\mathrm ...

\begin{aligned}{\frac {\mathrm {d} \Phi }{\mathrm {d} \varepsilon }}&={\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}L(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\,\mathrm {d} x\\&=\int _{a}^{b}{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}L(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))\,\mathrm {d} x\\&=\int _{a}^{b}\left[\eta (x){\frac {\partial L}{\partial {f}}}(x,f(x)+\varepsilon \eta (x),f'(x)+\varepsilon \eta '(x))+\eta ...