If `F(x)=|("f"_1,"f"_2,"f"_3),("g"_1,"g"_2,"g"_3),("h"_1,"h"_2,"h"_3)|` where `g_i,m f_i, h_i, (i=1, 2, 3)` are functions of `x`. To differentiate `F(x)`, we can differentiate one row (or column) at a time, keeping others unchanged.
`F'(x)=|("f'"_1,"f'"_2,"f'"_3),("g"_1,"g"_2,"g"_3),("h"_1,"h"_2,"h"_3)|+|("f"_1,"f"_2,"f"_3),("g'"_1,"g'"_2,"g'"_3),("h"_1,"h"_2,"h"_3)|+|("f"_1,"f"_2,"f"_3),("g"_1,"g"_2,"g"_3),("h'"_1,"h'"_2,"h'"_3)|`
`F'(x)=|("f'"_1,"f"_2,"f"_3),("g'"_1,"g"_2,"g"_3),("h'"_1,"h"_2,"h"_3)|+|("f"_1,"f'"_2,"f"_3),("g"_1,"g'"_2,"g"_3),("h"_1,"h'"_2,"h"_3)|+|("f"_1,"f"_2,"f'"_3),("g"_1,"g"_2,"g'"_3),("h"_1,"h"_2,"h'"_3)|`.
(1) If `F(x)=|(1,2a,3a^2),(x,x^2,x^3),(e^(x-a),e^(x^2-a^2),e^(x^3-a^3))|` then the value of `F'(a)` will be:
(a) `0`
(b) `1`
(c) `2`
(d) none of these

(2) If `F(x)=|(1,2a,3a^2),(x,x^2,x^3),(e^(x-a),e^(x^2-a^2),e^(x^3-a^3))|` then the value of `F'(a)` will be:
(a) `0`
(b) `1`
(c) `2`
(d) none of these

Answer

(1) A, (2) A