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)`

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))

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) .

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))

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

If f(x)=(a x^2+b)^3, then find the function g such that f(g(x))=g(f(x))dot

If f (a) = 2 , f'(a) = 1, g (a) = -1 and g'(a) = 2 , show that lim_(x to a)(g(x)f(a)-f(x)g(a))/(x-a) = 5

If f is a twice differentiable function such that f''(x)=-f(x),f'(x)=g(x) ,h(x)=[f(x)]^2+[g(x)]^2 if h(5)=11, then h(10) equals

Let g(x) be the inverse of an invertible function f(x), which is differentiable for all real xdot Then g^('')(f(x)) equals. (a) -(f^('')(x))/((f^'(x))^3) (b) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^3)/(f^(prime)(x)) (c) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^2)/((f^(prime)(x))^2) (d) none of these

If f(x),g(x)a n dh(x) are three polynomial of degree 2, then prove that phi(x)=|[f(x)g(x)h(x)];[f'(x)g'(x) h '(x)];[f' '(x)g' '(x )h ' '(x)]| is: constant

f(x) and g(x) are two differentiable functions on [0, 2] such that f''(x)-g''(x)=0, f'(1)=2, g'(1)=4, f(2)=3 and g(2)=9 , then [f(x)-g(x)] at x=(3)/(2) is equal to -