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if =|{:(x^(n),,n!,,2),(cos x,,cos.(npi)/...

if =`|{:(x^(n),,n!,,2),(cos x,,cos.(npi)/(2),,4),(sin x,,sin .(npi)/(2),,8):}|, then find the value of
(d^(n))/(dx^(n)) [f(x)]_(x=0) .(n in z)`.

Text Solution

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` (d^n)/(dx^n)[f(x)]= |{:( (d^(n))/(dx^(n))(x^(n)),,n!,,2),((d^(n))/(dx^(n))(cos x),,cos.(npi)/(2),,4),((d^(n))/(dx^(n)) (sinx),,sin .(npi)/(2),,8):}|`
`=|{:(n!,,n!,,2),(cos (x+(npi)/(2)),,cos .(npi)/(2),,4),(sin (x+(npi)/(2)),,sin .(npi)/(2),,8):}|( n in Z)`
`rArr (d^(n))/(dx^(n)) [f(x)]_(x=0) =0" "(:' C_(1) " and " C_(2) " are same)"`
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