If `f(x)=prod_(n=1)^(100)(x-n)^(n(101-n)` then find `(f(101))/(f '(101))`

If (d)/(dx)[ x^(n+1)+c]=(n+1)x^(n) , then find int x^(n)dx .

If f(x)= x^(n-1)+x^(n-2)+….+1,-1ltxlt1 then f'(x)=…….

If f(x)=(x^(4)+x^(2)+1)/(x^(2)-x+1) , the value of f(omega^(n)) (where 'omega' is the non-real root of the equation z^(3)=1 and 'n' is a multiple of 3), is ……….. .

If f(x) satisfies the relation, f(x+y)=f(x)+f(y) for all x,y in R and f(1)=5, then find sum_(n=1)^(m)f(n) . Also, prove that f(x) is odd.

f:N rarrN , f(x)=x+(-1)^(x-1) then f^(-1)(x)= .......

If d/(dx) (f(x))^(n)=n(f(x))^(n-1)(df(x))/(dx) then (d)/(dx) (sin^3x)=3sin^(2)x.cosx .

If f(x)=(a-x^(n))^(1/n) , where a gt 0 and n in N , then fof (x) is equal to

f: Z rarr Z, f(n)= (-1)^(n) . Find the range of f.

Let f:N rarr R be such that f(1)=1 and f(1)+2f(2)+3f(3)+…+nf(n)=n(n+1)f(n), for n ge 2, then (2010f(2010)) is ……….. .

Evaluation of definite integrals by subsitiution and properties of its : If I_(n)=int_(pi//4)^(pi//2)cot^(n)xdx, then 100(I_(99)+I_(101))=.......