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

Solve x dy=(y+x(f(y/x))/(f^(prime)(y/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))

Show that inte^x(f(x)+f^(prime)(x))dx=e^xf(x)+C

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

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-y), f(x) f(y) and f(x+y) are in A.P. for all x , y ,and f(0)!=0, then (a) f(4)=f(-4) (b) f(2)+f(-2)=0 (c) f^(prime)(4)+f^(prime)(-4)=0 (d) f^(prime)(2)=f^(prime)(-2)

Let f: R->R satisfying |f(x)|lt=x^2,AAx in R be differentiable at x=0. Then find f^(prime)(0)dot

Let y=f(x) be a parabola, having its axis parallel to the y-axis, which is touched by the line y=x at x=1. Then, (a) 2f(0)=1-f^(prime)(0) (b) f(0)+f^(prime)(0)+f(0)=1 (c) f^(prime)(1)=1 (d) f^(prime)(0)=f^(prime)(1)

Let f(x)=|x-1|dot Then (a) f(x^2)=(f(x))^2 (b) f(x+y)=f(x)+f(y) (c) f(|x|)=|f(x)| (d) none of these