Let h(x) be differentiable for all x and let `f(x) = (kx=e^(x)) h(x) where k is some constant. If h(0)=5, `h^(')(0) = -2` and `f^(')(0) =18` then the value of k is equal to

Let f(x) be a differentiable function such that f(x)=x^2 +int_0^x e^-t f(x-t) dt then int_0^1 f(x) dx=

Let f be a differentiable function such that f(1) = 2 and f'(x) = f (x) for all x in R . If h(x)=f(f(x)), then h'(1) is equal to

Suppose that f(0)=0a n df^(prime)(0)=2, and let g(x)=f(-x+f(f(x)))dot The value of g' (0) is equal to _____

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then

Let f(x) be defined for all x > 0 and be continuous. Let f(x) satisfies f(x/y)=f(x)-f(y) for all x,y and f(e)=1. Then

Let f(x) = (sec x)^(cosecx) + (cot x)^(sin x) . Then the value of Lim_(x rarr 0^(+)) f(x) is equal to

If x<0,t h e ntan^(-1)x is equal to

Let f, g and h are differentiable functions. If f(0) = 1; g(0) = 2; h(0) = 3 and the derivatives of theirpair wise products at x=0 are (fg)'(0)=6;(g h)' (0) = 4 and (h f)' (0)=5 then compute the value of (fgh)'(0) .

Let f be a differentiable function satisfying int_(0)^(f(x))f^(-1)(t)dt-int_(0)^(x)(cost-f(t)dt=0 and f((pi)/2)=2/(pi) The value of lim_(x to 0)(cosx)/(f(x)) is equal to where [.] denotes greatest integer function