I have the function : $$g(x,y)= f'_{x}(x,y)e^{f(x,y)-5}\sin y$$
I am trying to make a Taylor expansion and I am a little bit confused how to derive the function with respect to y. (I understand how it works with 2 functions) but when it comes to 3 I have no idea how to approach this kind of problem.
The product rule for differentiation can be generalized to three single-variable functions: $$ (fgh)' = f(gh)' + f'(gh) = f(g'h + gh') + f'gh = f'gh + fg'h + fgh'. $$ For partial differentiation of the product of two-variable functions $f$, $g$, and $h$, this rule becomes $(fgh)_y = f_ygh + fg_yh + fgh_y$.