Let `f(x)=|[cos(x+x^2),sin(x+x^2),-cos(x+x^2)],[sin(x-x^2),cos(x-x^2), sin(x-x^2)],[sin2x,0,sin(2x^2)]|` . Find the value of `f^(prime)(0)`.

Explanation

`f(x) =Delta (R_(1),R_(2),R_(3))+Delta(R_(1),R_(2),R_(3))+Delta(R_(1),R_(2),R_(3))`
For `x=0 ,Delta (R_(1),R_(2),R_(3))=0" and "Delta (R_(1),R_(2),R_(3))=0` as in both all the elements in `R_(3)` are zero Hence.
`f(0)=Delta(R_(1),R_(2),R_(3))_(x=0)`
`|{:(cos(x+x^(2)),,sin (x+x^(2)),,-cos (x+x^(2))),(sin (x-x^(2)),,cos(x-x^(2)),,sin (x-x^(2))),(2 cos 2x,,0,,4x cos (2x^(2))):}|`
`= |{:(1,, 0,,-1),(0,,1,,0),(2,,0,,0):}|=2`
Hence f(0)=2