If `g` is the inverse function of `fa n df^(prime)(x)=sinx ,t h e ng^(prime)(x)` is (a)`cos e c{g(x)}` (b) `"sin"{g(x)}` (c)`-1/("sin"{g(x)})` (d) none of these

If f(x)=x+tanxa n df is the inverse of g, then g^(prime)(x) equals (a) 1/(1+[g(x)-x]^2) (b) 1/(2-[g(x)-x]^2) (c) 1/(2+[g(x)-x]^2) (d) none of these

If u=f(x^3),v=g(x^2),f^(prime)(x)=cosx ,a n dg^(prime)(x)=sinx ,t h e n(d u)/(d v) is

If f(x)=|x-2|a n dg(x)=f[f(x)],t h e ng^(prime)(x)= ______ for x>2

Let g(x) be the inverse of an invertible function f(x) which is differentiable at x=c . Then g^(prime)(f(x)) equal. (a) f^(prime)(c) (b) 1/(f^(prime)(c)) (c) f(c) (d) none of these

If f(x)=(log)_x(lnx),t h e nf^(prime)(x) at x=e is equal to 1/e (b) e (c) 1 (d) zero

If f(x)=(log)_x(lnx),t h e nf^(prime)(x) at x=e is equal to 1/e (b) e (c) 1 (d) zero

Ifg(x)=int_0^x(|sint|+|cost|)dt ,t h e ng(x+(pin)/2) is equal to, where n in N , g(x)+g(pi) (b) g(x)+g((npi)/(n2)) g(x)+g(pi/2) (d) none of these

Suppose that f(x) is differentiable invertible function f^(prime)(x)!=0a n dh^(prime)(x)=f(x)dot Given that f(1)=f^(prime)(1)=1,h(1)=0 and g(x) is inverse of f(x) . Let G(x)=x^2g(x)-x h(g(x))AAx in Rdot Which of the following is/are correct? G^(prime)(1)=2 b. G^(prime)(1)=3 c. G^(1)=2 d. G^(1)=3

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)=0fora 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

Evaluate: ifintg(x)dx=g(x),t h e nintg(x){f(x)+f^(prime)(x)}dx