How would I find the partial derivatives of a function $$L(x(t), v(t), t)$$ where x and v are a function of time? I'm trying to derive the Euler-Lagrangian equation, and finding partial derivatives for $$x(t)+s \eta(t)$$ $$v(t)+s \eta(t)$$ of this expression:
$$ \frac{d}{ds} \int_{a}^{b} L(x(t)+s \eta(t),v(t)+s \eta(t),t) \, dt $$
seems too complex at the moment. Any hints? I know that in general, for $$L(x(t),y(t),z(t))$$ $$ \frac{dL}{dt}=\frac{dL}{dx}\frac{dx}{dt}+\frac{dL}{dy}\frac{dy}{dt}+\frac{dL}{dz}\frac{dz}{dt} $$
But how do I apply that to this problem?