If `f(x)=g(u) `and `u=u(x) `then
(A)`f'(x)=g'(u)`
(B)`f'(x)=g'(u)u'(x)`
(C)`f'(x)=u'(x)`
(D)None of these

If y=f(x) and x=g(y) are inverse of each other. Then g'(y) and g"(y) in terms of derivative of f(y) is A. -(f"(x))/([f'(x)]^(3)) B. (f"(x))/([f'(x)]^(2)) C. -(f"(x))/([f'(x)]^(2)) D. None of these

If f(x)=ax+b and g(x)=cx+d, then f(g(x))=g(f(x)) is equivalent to (A) f(a) = g(c ) (B) f(b) = g(b) (C) f(d) = g(b) (D) f(c ) = g(a)

Let y= int_(u(x))^(y(x)) f (t) dt, let us define (dy)/(dx) as (dy)/(dx)=v'(x) f (v(x)) - u' (x) f(u(x)) and the equation of the tangent at (a,b) and y-b=((dy)/(dx))(a,b) (x-a) . If y=int_(x^(2))^(x^(4)) (In t) dt , "then" lim_(x to 0^(+)) (dy)/(dx) is equal to

If (x-1) is a factor of polynomial f(x) but not of g(x) , then it must be a factor of (a) f(x)g(x) (b) -f\ (x)+\ g(x) (c) f(x)-g(x) (d) {f(x)+g(x)}g(x)

If y = underset(u(x))overset(v(x))intf(t) dt , let us define (dy)/(dx) in a different manner as (dy)/(dx) = v'(x) f^(2)(v(x)) - u'(x) f^(2)(u(x)) alnd the equation of the tangent at (a,b) as y -b = (dy/dx)_((a,b)) (x-a) If F(x) = underset(1)overset(x)inte^(t^(2)//2)(1-t^(2))dt , then d/(dx) F(x) at x = 1 is

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

Given g(x)=(1/x), h(x)=x^(2)+2x+(lamda+1) and u(x)=1/x+cos(1/(x^(2))) Let f(x)=lim_(ntooo)(x^(2n+1)g(x)+h(x))/(x^(2n)+3x.u(x)) The value of lim_(xtooo)f(x) is

Given g(x)=(1/x), h(x)=x^(2)+2x+(lamda+1) and u(x)=1/x+cos(1/(x^(2))) Let f(x)=lim_(ntooo)(x^(2n+1)g(x)+h(x))/(x^(2n)+3x.u(x)) If lim+(xto2)f(x)=I , then [I] (where [.] denotes the greatest integer function), is equal to :

Given g(x)=(1/x), h(x)=x^(2)+2x+(lamda+1) and u(x)=1/x+cos(1/(x^(2))) Let f(x)=lim_(ntooo)(x^(2n+1)g(x)+h(x))/(x^(2n)+3x.u(x)) If lim_(xto0)f(x)=2 , then the value of lamda is

Let u(x) and v(x) satisfy the differential equation (d u)/(dx)+p(x)u=f(x) and (d v)/(dx)+p(x)v=g(x) are continuous functions. If u(x_1) for some x_1 and f(x)>g(x) for all x > x_1, prove that any point (x , y), where x > x_1, does not satisfy the equations y=u(x) and y=v(x)dot