Going through a paper recently I got stuck on the simple differential analysis that the authors were using. I had not come across this before, so maybe there is an elegant way to explain this.
In a 2018 paper on solar photovoltaics (P.9 Main Body, P1 in Supplementary Material), the authors have a cost function $C$ which describes the cost associated with manufacturing one unit and depends on manufacturing variables $x,y$, which change over time (e.g. price of silicon, price of chemicals, etc.)
$$
C(x(t),y(t))
$$
They want to determine the contribution of a single variable $x$ to the total change of the cost function between two points in time $Delta C (t_0, t_1)$. Variables are known only at discrete points in time ($t_0,t_1$).
They start by writing out the differential of the cost function $C$ as
$$
dC (x(t), y(t)) = frac{ partial C }{ partial x } frac{ text{d} x }{ text{d} t} text{d} t + frac{ partial C }{ partial y } frac{ text{d} y }{ text{d} t} text{d} t
$$
where the contribution of the change in variable x over time $t_0 < t < t_1$ is then
$$
Delta C_x = int_{t=t_0}^{t_1} frac{ partial C }{ partial x } frac{ text{d} x }{ text{d} t} text{d} t
$$
Here they say
If it were possible to observe the (…) variables x in continuous time, (…) (this equation) would provide all that is needed to compute the contribution of each variable x.
Using logarithmic differentiation, they go on to rewrite the expression as
$$
Delta C_x = int_{t=t_0}^{t_1} C(t) frac{ partial ln C }{ partial x } frac{ text{d} x }{ text{d} t} text{d} t
$$
and then for $C(t)$ assume a constant $C(t) approx tilde{C} $ which is ultimately chosen to be $tilde{C} = frac{ Delta tilde{C} }{ Delta ln tilde{C} }$, such that $Delta C_x + Delta C_y = Delta C$.
My questions:

Why not assume all variables $x,y$ change linearly between $t_0$ and $t_1$, and then integrate:
$$
Delta C_x = int_{t=t_0}^{t_1} frac{ partial C }{ partial x } frac{ text{d} x }{ text{d} t} text{d} t
$$ 
Even if the time dependence of variables was known (eg. daily data on the price of silicon, etc.), then integrating would not yield what the authors are actually looking for. They are interested in the contribution of single variables to the total change in $C$ (eg. what percentage of total manufacturing cost reductions are due to decrease in silicon price). But integrating using $Delta C_x = int_{t=t_0}^{t_1} frac{ partial C }{ partial x } frac{ text{d} x }{ text{d} t} text{d} t $
would yield different results for different time dependency of variables. A variable $x(t)$ (purple)
would yield a different $Delta C_x$ than a variable $x'(t)$ (blue).