I am exploring Taylor's Classical Mechanics and I am unsure about the mathematics when he derives the minimum of the action. We define this function $f(x(t),\dot{x}(t),t)$ to be a Lagrangian. We define the action to be $$ S = \int_{t_{1}}^{t_{2}} f(x(t),\dot{x}(t),t)dt$$
Let some function $x$ satisfy the condition that $S$ is minimum. Then we define $X(t) = x(t)+\alpha \eta (t)$ to be a family of functions that $x$ lives in.
By design, we know that the minimum of $S$ occurs when $\alpha = 0$. We then go on the differentiate $S$ with respect to $\alpha$ where the integrand is $f(X(t),\dot{X}(t),t)$.
We get to a point where we are finding $$\frac{\partial f(x(t)+\alpha \eta(t),\dot{x}(t)+\alpha \dot{\eta}(t),t)}{\partial \alpha}$$ My understanding of the chain rule is that this result is supposed to be the partial of $f$ with respect to the first argument multiplied by the partial of the first argument with respect to $\alpha$ and so on but this is not reflected in the textbook. The result of the expression from the book is $\frac{\partial f}{\partial x}\eta(t)+\frac{\partial f}{\partial \dot{x}}\dot{\eta}(t)$. Shouldn't these $x$ be $X$?
Edit:
I guess wat I am asking is, why does the book use $\frac{\partial f}{\partial x}\eta(t)+\frac{\partial f}{\partial \dot{x}}\dot{\eta}(t)$ and not $\frac{\partial f}{\partial (x(t)+\alpha \eta(t))}\eta(t)+\frac{\partial f}{\partial (\dot{x}(t)+\alpha \dot{\eta}(t))}\dot{\eta}(t)$? Are these the same thing or is it a matter of convention?
From a certain perspective, you could think of it as a (relatively benign) "abuse of notation." If you view $\Xi \equiv (X,\dot X)$ simply as the symbols for variables that $f$ depends on (along with $t$), then to compute the full derivative $\frac{d}{d\alpha} f(x + \alpha \eta(t), \dot x + \alpha \dot \eta(t), t)$ using the chain rule, you would first write $f(x+\alpha\eta,\dot x+\alpha\dot\eta,t)$ as $f(\Xi(\alpha), t)$, where $\Xi(\alpha) \equiv (x(t) + \alpha \eta(t), \dot x(t) + \alpha \dot\eta(t))$ and then in 'Shifrin-esque' notation) \begin{align*} D_\alpha f(\Xi(\alpha), t) &= D_{\Xi} f D_{\alpha}\Xi,\\ &=\partial_Xf \frac{d}{d\alpha} X + \partial_{\dot X}f\frac{d}{d\alpha}\dot X\\ &=\partial_X f \frac{d}{d\alpha}(x(t) + \alpha \eta(t)) + \partial_{\dot X} f\frac{d}{d\alpha}(\dot x(t) + \alpha \dot \eta(t))\\ &= \partial_X f \eta(t) + \partial_{\dot X} f\dot \eta(t) \end{align*} You then evaluate $\partial_Xf$ and $\partial_{\dot X} f$ at $\alpha = 0$, which produces the "simplified" (though possibly unclear from abuse of notation) expression, \begin{align*} \partial_xf(x,\dot x,t)\eta(t) + \partial_{\dot x}f(x,\dot x, t)\dot \eta(t), \end{align*} integrate by parts, yyy..