Using the first principle, prove that `d/(dx)(1/(f(x)))=(-f^(prime)(x))/([f(x)]^2)`

Using the first principle, prove that: d/(dx)(f(x)g(x))=f(x)d/(dx)(g(x))+g(x)d/(dx)(f(x))

Suppose fa n dg are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all xa n df^(prime)a n dg' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x))e q u a l (a) (-2f^(prime)(x))/f (b) (2g^(prime)(x))/(g(x)) (c) (-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))

Let the function ln f(x) is defined where f(x) exists for x geq 2 and k is fixed positive real numbers prove that if d/(dx) (x.f(x)) geq -k f(x) then f(x) geq Ax^(-1-k) where A is independent of x.

The function y=f(x) is the solution of the differential equation (dy)/(dx)+(x y)/(x^2-1)=(x^4+2x)/(sqrt(1-x^2)) in (-1,1) satisfying f(0)=0. Then int_((sqrt(3))/2)^((sqrt(3))/2)f(x)dx is

Let f(x) be differentiable function and g(x) be twice differentiable function. Zeros of f(x),g^(prime)(x) be a , b , respectively, (a

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

Find the derivative of x from the first principle, where f is given by f(x)=x+(1)/(x)

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

Given f^(prime)(1)=1"and"d/(dx)(f(2x))=f^(prime)(x)AAx > 0 .If f^(prime)(x) is differentiable then there exies a number c in (2,4) such that f''(c) equals

If function f satisfies the relation f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) for all x ,