A function g defined for all positive real numbers, satisfies `g'(x^2) =x^3` for all `xgt0` and `g(1)=1` Compute g (4).

A function g defined for all positive real numbers,satisfies g'(x^(2))=x^(3) for all x>0 and g(1)=1 compute g(4)

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'

let f(x)=sqrt(x) and g(x)=x be function defined over the set of non negative real numbers. find (f+g)(x),(f-g)(x),(fg)(x) and (f/g)(x)

Let f(x)=sqrt(x) and g(x)" "=" "x be two functions defined over the set of nonnegative real numbers. Find (f" "+" "g)" "(x) , (f" "" "g)" "(x) , (fg)" "(x) and (f/g)(x) .

Let the function f, g, h are defined from the set of real numbers RR to RR such that f(x) = x^2-1, g(x) = sqrt(x^2+1), h(x) = {(0, if x lt 0), (x, if x gt= 0):}. Then h o (f o g)(x) is defined by

Let f(x) and g(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))

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^(y)g(y) for all x and y then:

Consider f g and h are real-valued functions defined on R. Let f(x)-f(-x)=0 for all x in R, g(x) + g(-x)=0 for all x in R and h (x) + h(-x)=0 for all x in R. Also, f(1) = 0,f(4) = 2, f(3) = 6, g(1)=1, g(2)=4, g(3)=5, and h(1)=2, h(3)=5, h(6) = 3 The value of f(g(h(1)))+g(h(f(-3)))+h(f(g(-1))) is equal to

If a function g(x) which has derivaties g'(x) for every real x and which satisfies the following equation g(x+y) = e^(y)g(x) + e^(x)g(x) for all x and y and g'(0) = 2, then the value of {g'(x) - g(x)} is equal to

Let f and g be differrntiable functions on R (the set of all real numbers) such that g (1)=2=g '(1) and f'(0) =4. If h (x)= f (2xg (x)+ cos pi x-3) then h'(1) is equal to: