# Chain rule or Substitution rule?

Chain rule or Substitution rule???

## Homework Statement

It appears that a standard result in ODE is the following: if $f(x,t)$ is smooth enough, then the solution $\psi(t)$ to the initial value problem:

$x(t)=x_{0}+\int_{0}^{t}f(x(s),s)ds$
$x(0)=x_{0}$

is continuously differentiable with respect to its initial condition (or a parameter in the problem), and that this derivative satisfies the following first variational equation:

$\frac{\partial \psi(t)}{\partial x}=c +\int_{0}^{t}\frac{\partial f(\psi(s,x),s)}{\partial x}\frac{\partial \psi}{\partial x}ds$

## Homework Equations

To obtain the second equation above one makes use of the chain rule applied to $f(\psi(t,x),t)$. However, I am not certain we can always use this rule. Consider the following `f' function:

$f(\psi(s),s)=\biggl(\frac{1-\int_{0}^{s}G(\psi(\xi))d\xi}{F(t)F(\psi(t))+\int_{0}^{t}G(\psi(\xi))d\xi}\biggl)^N$

where $G$ is a sufficiently continuously differentiable function, and $N>2$. The question is how could I apply the chain rule to obtain a first variational equation as the one I presented above?

## The Attempt at a Solution

No sure yet. I truly need help here as I've trying to understand this for a long time without any success...Thanks!