Consider the following integral
$$I = \int^b_af[x[t];c]dt$$
where $a,b$ and $c$ are scalers and $x[t]$ is a function of $t$. I want to know $\frac{\partial I}{\partial x}$.
If I apply Leibniz Rule, I get
$$\frac{\partial I}{\partial x} = \int^b_a\frac{\partial f}{\partial x}dt$$
I feel this is an incorrect application of the rule, since $x$ is a function of $t$.
Alternatively, I can think of re-defining $f$ as $g = f[x+\epsilon;c]$, where for $\epsilon = 0$ we have the original function. Now if I apply Leibniz Rule by differentiating with respect to $\epsilon$ (and then evaluate the derivative at $\epsilon = 0$), I get,
$$\frac{\partial I}{\partial \epsilon} = \int^b_a\frac{\partial g}{\partial x}|_{\epsilon=0} dt$$
Questions:
- Am I correct in saying that my first method is incorrect?
- If the alternative method is correct, does it correctly capture $\frac{\partial I}{\partial x}$?
Edit:
It might help to explain what I was thinking of when I wrote the question:
Suppose $x$ represents monetary values at different points of time ($t$) and $f$ is a function of $x$. $I$ is adding $f$ from $t=a$ to $t=b$. I would like to calculate, how the sum ($I$) changes if the entire sequence $x$, say, goes up or down.