Let `f(x)a n dg(x)` be two function having finite nonzero third-order derivatives `f'''(x)` and `g'''(x)` for all `x in R`. If `f(x).g(x)=1` for all `x in R ,` then prove that `(f''')/(f') - (g''')/(g') = 3((f'')/f - (g'')/g)`.

IF f(x)g(x)=a, for all x [a is a constant], show that, (g''(x))/(g'(x))=(f''(x))/(f'(x))-(2f'(x))/f(x)

If g is inverse of f then prove that f''(g(x))=-g''(x)(f'(g(x)))^(3).

Let f(x) and g(x) be real valued functions such that f(x)g(x)=1, AA x in R."If "f''(x) and g''(x)" exists"AA x in R and f'(x) and g'(x) are never zero, then prove that (f''(x))/(f'(x))-(g''(x))/(g'(x))=(2f'(x))/(f(x))

Suppose f and g are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all x and f^(prime) and g' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x)) is equal (a) (-2f^(prime)(x))/f (b) (-2g^(prime)(x))/(g(x)) (c) (-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))

Let f(x)=sqrtx and g(x)=x be two functions defined over the set of non-negative real numbers. Find (f+g) (x), (f-g), (fg) (x) and (f/g) (x) .

If f(x) and g(x) be two given function with all real numbers as their domain, then h(x)={f(x)+f(-x)}{g(x)-g(-x)} is

If f(x)=log_ex and g(x)=e^x , prove that f{g(x)}=g{f(x)} .

Let f,g: R-R be two functions defined as f(x)=|x|+x and g(x)=|x|-x , for all x Then find fog and gof.

Let F : R to R be such that f is injective and f(x) f(y) =f(x+y) for all x,y in RR If f(x) , f(y) , f(z) are in G.P ., then x,y,z are in -

Let f(x) = x + cos x + 2 and g(x) be the inverse function of f(x), then g'(3) equals _