If `f(x)=|(x^n, sinx, cosx),(n!, sin((npi)/2), cos((npi)/2)),(a, a^2,a^3)|`, then show that `d^n/dx^n [f(x)]` at x=0` is 0

If f(x)=|{:(x^(n),sinx,cosx),(n!,"sin"(npi)/(2),"cos"(npi)/(2)),(a,a^(2),a^(3)):}| , then the value of (d^(n))/(dx^(n))(f(x))" at "x=0" for "n=2m+1 is

Let "f(x)"=|{:(pi^n,sinpix,cospix),((-1)^(n)!,-sin((npi)/2),-cos((npi)/2)),(-1,1/sqrt2,sqrt3/2):}| Then value or d^n/(dx^n)["f(x)"]"at "x=1" is "

If f(x)=|[x^n, n!, 2; cosx, cos((npi)/2), 4; sinx ,sin((npi)/2), 8]| then find the value of (d^n)/(dx^n)([f(x)])_(x=0)dot(n in z)dot

If f(x)=|x^nn !2cosxcos(npi)/2 4sinxsin(npi)/2 8| then find the value of (d^n)/(dx^n)([f(x)])_(x=0)dot(n in z)dot

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).

Let f(x) = |(2cos^2x, sin2x, -sinx), (sin2x, 2 sin^2x, cosx), (sinx, -cosx,0)| , then the value of int_0^(pi//2){f(x) + f'(x)} dx is

If int_(0)^(npi) f(cos^(2)x)dx=k int_(0)^(pi) f(cos^(2)x)dx , then the value of k, is

Let f(x)=lim_(n to oo) ((2 sin x)^(2n))/(3^(n)-(2 cos x)^(2n)), n in Z . Then

If x ne (npi)/2, n in I and (cosx)^(sin^(2)x-3sinx+2)=1 , then find the general solutions of x.

f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3). Then the least value of n for which (d^(n))/(dx^(n))f(x)|underset(x=0) is nonzero is