I can't solve or understand question involving the chain rule with two variables.

Question

The question wants me to find the differential at any point $(x,t)$ for $f(x,t)=e^{−4t}sin(x+3t)$ and I just can't understand this at all. It feels like it lacks information. I would really appreciate if someone could explain me how I'm supposed to solve this. (Along with an answer, if possible.) Thanks in advance.

Answer 1

The differential $df$ of $f$ can be expressed in terms of the coordinate differentials $dx$, $dt$ as $$df(x,t)=f_x(x,t)dx+f_t(x,t)dt\ ,$$ whereby $f_x$, $f_t$ denote the partial derivatives of $f$ with respect to the indicated variable. But this is just legalese; it only aquires meaning if you have an idea of what a "differential of a function" is.