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

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

Let g(x) be the inverse of an invertible function f(x) which is derivable at x=3 . If f(3)=9 and f^(prime)(3)=9 , write the value of g^(prime)(9) .

If differentiable function f(x) in inverse of g(x) then f^(g(x)) is equal to (g^(x))/((g^(prime)(x))^3) 2. 0 3. (g^(x))/((g^(prime)(x))^2) 4. 1

int \ {f(x)*g^(prime)(x)-f^(prime)(x)g(x))/(f(x)*g(x)){logg(x)-logf(x)} \ dx

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 f(x)=|x-a|varphi(x), where varphi(x) is continuous function, then (a) f^(prime)(a^+)=varphi(a) (b) f^(prime)(a^-)=-varphi(a) (c) f^(prime)(a^+)=f^(prime)(a^-) (d) none of these

If f^(prime)(x)=8x^3-2x ,f(2)=8, find f(x)

Using first principles, prove that d/(dx){1/(f(x))}=-(f^(prime)(x))/({f(x)}^2)

If f(a)=2,f^(prime)(a)=1,g(a)=-1,g^(prime)(a)=2, then the value of lim_(x->a)(g(x)f(a)-g(a)f(x))/(x-a) is (a) -5 (b) 1/5 (c) 5 (d) none of these

If f(x)=sqrt(x^2+1),\ \ g(x)=(x+1)/(x^2+1) and h(x)=2x-3 , then find f^(prime)(h^(prime)(g^(prime)(x))) .