Let f: `R → R` be a one-one onto differentiable function, such that `f(2)=1 and f^(prime)(2)=3.` Then, find the value of `(d/(dx)(f^(-1)(x)))_(x=1)`

The value of (d)/(dx)5^(f(x)) is -

Let f:[(1)/(2),1]to RR (the set of all real numbers) be a positive, non-constant and differentiable function such that f'(x)lt2f(x)" and "f((1)/(2))=1 Then the value of int_((1)/(2))^(1)f(x)dx lies in the interval

Let f:R to R and h:R to R be differentiable functions such that f(x)=x^(3)+3x+2,g(f(x))=x and h(g(g(x)))=x for all x in R . Then, find the value of h'(1).

Let f:R to R be a differentiable function and f(1)=4 and f'(1)=2 , then the value of lim_(x to 1) int_(4)^(f(x))(2t)/(x-1)dt is -

Let f:[1/2,1]->R (the set of all real numbers) be a positive, non-constant, and differentiable function such that f^(prime)(x)<2f(x) and f(1/2)=1 . Then the value of int f(x)dx lies in the interval for x:[1/2,1] (a) (2e-1,2e) (b) (3-1,2e-1) (c) ((e-1)/2,e-1) (d) (0,(e-1)/2)

Let f: R->R be a differentiable function having f(2)=6,f^(prime)(2)=1/(48)dot Then evaluate ("lim")_(x->2)int_6^(f(x))(4t^3)/(x-2)dt

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 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

If f(x)a n dg(x) are differentiable functions for 0lt=xlt=1 such that f(0)=10 ,g(0)=2,f(1)=2,g(1)=4, then in the interval (0,1)dot (a) f^(prime)(x)=0 for a l lx (b) f^(prime)(x)+4g^(prime)(x)=0 for at least one x (c) f(x)=2g'(x) for at most one x (d) none of these