Suppose I invent a nonstandard notation to clarify this. For the purposes of this question, I will stipulate an expansion of Leibniz's notation and denote it in [imath]\color{blue}{blue}[/imath]. The expansion is as follows: Expressions can appear in the "numerator" and/or "denominator" of ordinary and partial derivatives (i.e., if [imath]b = a^4[/imath], then [imath]\frac{dc}{db}[/imath] could be written as [imath]\color{blue}{\frac{dc}{d(a^4)}}[/imath]; additionally, a value in parentheses to be plugged in for a variable (or expression) after differentiation also includes the variable, in the form "variable [imath]\rightarrow[/imath] value" (i.e., [imath]\color{blue}{\frac{dc}{d(a^4)}(a^4 \rightarrow 2)}[/imath] more clearly states what might be meant by the more standard but ambiguous [imath]c'(2)[/imath]). Now onto the problem.
[math]\begin{cases}\sin(\pi x) = h(x) + g(-x) \\ 0 = \frac{\partial}{\partial t}[h(2t + x) + g(2t - x)]_{t = 0}\end{cases}[/math]
By the chain rule, the second equation becomes [imath]0 = 2\color{blue}{\frac{dh}{d(2t + x)}(t \rightarrow 0)} + 2\color{blue}{\frac{dg}{d(2t - x)}(t \rightarrow 0)}[/imath], easily rewritten as [imath]\color{blue}{\frac{dh}{d(2t + x)}(t \rightarrow 0)} + \color{blue}{\frac{dg}{d(2t - x)}(t \rightarrow 0)} = 0[/imath], but because the functions [imath]h(x,\ t)[/imath] and [imath]g(x,\ t)[/imath] are such that they only contain [imath]t[/imath] in the pattern of [imath]2t + x[/imath] and [imath]2t - x[/imath], respectively, this is the same as [imath]\color{blue}{\frac{dh}{d(2t + x)}((2t + x) \rightarrow x)} + \color{blue}{\frac{dg}{d(2t - x)}((2t - x) \rightarrow -x)} = 0[/imath]. This is ostensibly what bigjohn 2 means by the notation [imath]h'(x) + g'(-x) = 0[/imath].
The substitution [imath]\tilde g(x) = g(-x)[/imath] leads to the system
[math]\begin{cases}h(x) + \tilde g(x) = \sin(πx) \\ \color{blue}{\frac{dh}{d(2t + x)}((2t + x) \rightarrow x)} - \color{blue}{\frac{d\tilde g}{d(2t - x)}((2t - x) \rightarrow x)} = 0\end{cases}[/math]
The second equation now needs to be antidifferentiated, but I don't know with respect to what, so I have [imath]\int \color{blue}{\frac{dh}{d(2t + x)}((2t + x) \rightarrow x)} - \color{blue}{\frac{d\tilde g}{d(2t - x)}((2t - x) \rightarrow x)}\ d? = \int 0\ d?[/imath]. My first hypothesis was that I should antidifferentiate both sides with respect to [imath]t[/imath], producing an arbitrary function [imath]f(x)[/imath] on the RHS, because the equation was partially differentiated with respect to [imath]t[/imath] *initially*. I also considered antidifferentiating with respect to [imath]2t + x[/imath] or [imath]2t - x[/imath], because these are the expressions [imath]h[/imath] and [imath]\tilde g[/imath] are respectively differentiated with respect to *currently*, but the problem with such an approach is that I cannot antidifferentiate different terms in the equation with respect to different expressions. Finally, I could antidifferentiate with respect to [imath]x[/imath], producing an arbitrary function [imath]f(t)[/imath] on the RHS.
How should this equation be antidifferentiated? With respect to what variable or expression is the antiderivative, what is the rational behind this choice, and what is the solution to this antidifferentiated equation? Please do not rely on any understanding being conveyed by Lagrange's notation.