The method of finite differences will be used here. Consider a closed real interval $\closedint a b$, which is divided in $n + 1$ equal parts. Choose its subdivision to be normal: :$a = x_0 < x_1 < \cdots < x_n < x_{n + 1} = b$ such that for $i \in set {0, 1, \ldots, n - 1, n}$ we have $x_{i + 1} - x_i = \Delta x$. Approximate the desired function $y$ by a polygonal line with vertices $\tuple {x_i, y_i}$ where $i \in \set {0, 1, \ldots, n, n + 1}$, where $y_i = \map y {x_i}$. Hence, the functional $J \sqbrk y$ can be approximated by the following sum: :$\displaystyle \map {\mathscr J} {y_1, y_2, \ldots, y_{n - 1}, y_n} = \sum_{i \mathop = 0}^n \map F {x_i, y_i, \frac {y_{i + 1} - y_i} {\Delta x} } \Delta x$ Note that the values $\map y {x_0} = A$ and $\map y {x_1} = B$ are fixed, and therefore not varied. Now, consider a partial derivative of $J$ with respect to $y_k$, where $k \in \set {1, 2, \ldots, n - 1, n}$. {{begin-eqn}} {{eqn | l = \frac {\partial \mathscr J} {\partial y_k}       | r = \frac \partial {\partial y_k} \sum_{i \mathop = 0}^n \map F {x_i, y_i, \frac {y_{i + 1} - y_i} {\Delta x} } \Delta x       | c =  }} {{eqn | r = \sum_{i \mathop = 0}^n \frac \partial {\partial y_k} \map F {x_i, y_i, \frac {y_{i + 1} - y_i} {\Delta x} } \Delta x       | c =  }} {{eqn | r = \sum_{i \mathop = 0}^n \paren {\frac {\partial F} {\partial y_i} \paren {x_i, y_i, \frac {y_{i + 1} - y_i} {\Delta x} } \frac {\partial y_i} {\partial y_k} + \frac {\partial F} {\partial {\frac {y_{i + 1} - y_i} {\Delta x} } } \paren {x_i, y_i, \frac {y_{i + 1} - y_i} {\Delta x} } \frac {\partial {\frac {y_{i + 1} - y_i} {\Delta x} } } {\partial y_k} } \Delta x       | c =  }} {{end-eqn}} As all the functions $y_i$ are independent {{WRT}} each other, we have $\dfrac {\partial y_m} {\partial y_k} = \delta_{m k}$, where $\delta_{m k}$ is the Kronecker Delta. Then the aforementioned sum simplifies to: {{explain|"aforementioned" -- reference it directly, using a label}} :$\dfrac {\partial \mathscr J} {\partial y_k} = \paren {\dfrac {\partial F} {\partial y_k} \paren {x_k, y_k, \dfrac {y_{k + 1} - y_k} {\Delta x} } + \dfrac {\partial F} {\partial {\frac {y_k - y_{k - 1} } {\Delta x} } } \paren {x_{k - 1}, y_{k - 1}, \dfrac {y_k - y_{k - 1} } {\Delta x} } \dfrac 1 {\Delta x} - \dfrac {\partial F} {\partial {\frac {y_{k + 1} - y_k} {\Delta x} } } \paren {x_k, y_k, \dfrac {y_{k + 1} - y_k} {\Delta x} } \dfrac 1 {\Delta x} } \Delta x$ In order to get a variational derivative, the denominator of the {{LHS}} has to represent an area.  For this reason, divide everything by $\Delta x$, and take the limit $\Delta \to 0$. Then for all $k \in \set {1, 2, \dotsc, n - 1, n}$: {{begin-eqn}} {{eqn | l = \lim_{\Delta x \mathop \to 0} \frac {y_{k + 1} - y_k} {\Delta x}       | r = \lim_{\Delta x \mathop \to 0} \frac {\map y {x_{k + 1} } - \map y {x_k} } {\Delta x}       | c =  }} {{eqn | r = \lim_{\Delta x \mathop \to 0} \frac {\map y {x_k + \Delta x} - \map y {x_k} } {\Delta x}       | c =  }} {{eqn | r = \map {y'} {x_k}       | c = {{Defof|Derivative of Real Function at Point}} }} {{end-eqn}} Similarly, for $\map F {x, y, y'}$ we have {{begin-eqn}} {{eqn | r = \lim_{\Delta x \mathop \to 0} \frac {\map {F_{\map {y'} {x_k} } } {x_k, y_k, \map {y'} {x_k} } - \map {F_{\map {y'} {x_{k - 1} } } } {x_{k - 1}, y_{k - 1}, \map {y'} {x_{k - 1} } } } {\Delta x}       | o =        | c =  }} {{eqn | r = \lim_{\Delta x \mathop \to 0} \frac {\map {F_{\map {y'} {x_{k-1} + \Delta x} } } {x_{k - 1} + \Delta x, \map y {x_{k - 1} + \Delta x}, \map {y'} {x_{k - 1} + \Delta x} } - \map {F_{\map {y'} {x_{k - 1} } } } {x_{k-1}, \map y {x_{k - 1} }, \map {y'} {x_{k - 1} } } } {\Delta x}       | c =  }} {{eqn | r = \frac \d {\d x} \map {F_{\map {y'} {x_{k - 1} } } } {x_{k - 1}, \map y {x_{k - 1} }, \map {y'} {x_{k - 1} } } }} {{end-eqn}} Thus: :$\displaystyle \lim_{\Delta x \mathop \to 0} \frac {\partial \mathscr J} {\partial y_k \Delta x} = \map {F_{\map y {x_k} } } {x_k, \map y {x_k}, \map {y'} {x_k} } - \frac \d {\d x} \map {F_{\map {y'} {x_{k - 1} } } } {x_{k - 1}, \map y {x_{k - 1} }, \map {y'} {x_{k - 1} } }$ Note that the denominator on the left is an area covered by a rectangle with sides $\Delta x$ and $\partial y$, and vanishes as $\Delta x \to 0$. Finally, since the distance between any two neighbouring points approaches 0 as $\Delta x \to 0$, the set of all $x_k \in \closedint a b$ can be treated as continuous, and the index $k$ dropped: :$\displaystyle \lim_{\Delta x \mathop \to 0} \frac {\partial J} {\partial y \Delta x} = \map {F_{\map y x} } {x, \map y x, \map {y'} x} - \frac \d {\d x} \map {F_{\map {y'} x} } {x, \map y x, \map {y'} x}$ The {{LHS}} by definition is a variational derivative.  Suppose the {{LHS}} vanishes. Then the {{RHS}} vanishes as well. {{finish|Refine vanishing area with $\partial y$-probably introduce grid for $y$ as well; make the proof of set of $x_k$ becoming a real line more rigorous}}