I have a line $L\subset\mathbb{C}^n$ which is parametrized by $x_1=a_1t, x_2=a_2t,\dots, x_n=a_nt$, a function $f(x_1,\dots,x_n)$, and I want to look at the restriction of $f$ onto $L$. This is just $f(a_1t,\dots,a_nt)$, but the part I'm having trouble with is that I want an expression for $\partial f/\partial x_i$ restricted to L, i.e. as a function of $t$.
Is there some simple way to do this with the chain rule? Of course simply taking $\partial f/\partial x_i = dt/dx_i \ df/dt$ doesn't work. Is there any simple expression for the derivative that does not directly involve $\partial f/\partial x_i$?