Given a function ' g' which has a derivative `g' (x)` for every real x and satisfies `g'(0)=2 and g(x+y)=e^y g(x)+e^x g(y)` for all x and y then: Find g(x).

f(x)=e^x.g(x) , g(0)=2 and g'(0)=1 then f'(0)=

Let f be a function such that f(x+y)=f(x)+f(y) for all x and y and f(x)=(2x^2+3x)g(x) for all x , where g(x) is continuous and g(0)=3. Then find f^(prime)(x)dot

If f(x)=e^x and g(x) =log_(e)x, then find (f+g)(1) and fg(1).

If the function f:RR to RR is given by f(x) = x for all x inRR and the function g:RR-{0} to RR is given by g(x)=(1/x), "for all "x in RR-{0}, then find the function f+g and f-g.

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) .

If y=g{g(x)}, g(0)=0 and g'(0)=2 then find dy/dx at x=0

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))

Let f(x) be a function satisfying f'(x)=f(x) with f(0)=1 and g(x) be a function that satisfies f(x)+g(x)=x^(2) . Then the value of the integral int_(0)^(1) f(x)g(x) dx is equal to -

Let f : R rarr R be any function. Define g : R rarr R by g(x)=|f(x)| for all x. Then g is:

A twice differentiable function f(x)is defined for all real numbers and satisfies the following conditions f(0) = 2; f'(0)--5 and f '' (0) = 3 . The function g(x) is defined by g(x) = e^(ax) + f (x) AA x in R , where 'a' is any constant If g'(0) + g''(0)=0 . Find the value(s) of 'a'