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" Let " =|{:(cos x,,sin x,,cosx),( cos...

`" Let " =|{:(cos x,,sin x,,cosx),( cos 2x,,sin 2x,,2cos 2x),(cos 3x,,sin 3x,,3cos 3x):}|` then find the values of `f(0)` and f'`(pi//2)`.

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`f(x) =|{:(cos x,,sin x,,cosx),( cos 2x,,sin 2x,,2cos 2x),(cos 3x,,sin 3x,,3cos 3x):}|`
Differending column wise with respecte to x we get
`f(x) |{:(cos x ,,sin x,,cos x),(-2 sin x,,sin2x,,2 cos 2x),(cos x,,sin 3x,,3cos 3x):}|`
`+|{:(cos x,,cos x,,cos 3x ),(cos 2x,,2cos 2x,,2cos 2x),(cos x,,3cos 3x,,3cos 3x):}|`
`=|{:(-sin x,,sin x,,cos x),(-2 sin 2x,,sin 2x,,2 cos 2x),(=3 sin 3x,,sin 3x,, -9 sin 3x):}|`
`+|{:(cos x,,sin x ,,-sin x),(cos 2x,,sin 2x,,-4 sin x),(cos 3x,,sin 3x,,-9sinx):}|`
`f'(0) = |{:(0,,0,,1),(0,,0,,2),(0,,0,,3):}|+|{:(1,,0,,0),(1,,0,,0),(1,,0,,0):}|=0+=0`
`f(pi//2) = |{:(-1,,1,,0),(0,,0,,-2),(3,,1,,0):}| +|{:(0,,1,,-1),(-1,,0,,0),(0,,-1,,0):}|` a
`=(-6 +2) +(-1+9)`
`=4`
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