`f^(prime)(x)=phi(x)` and `phi^(prime)(x)=f(x)` for all x.Also,f(3)=5 and `f^(prime)(3)=4`. Then the value of `[f(10)]^2-[phi(10)]^2` is _____

If function f satisfies the relation f(x)xf^(prime)(-x)=f(-x)xf^(prime)(x)fora l lx ,a n df(0)=3,a n diff(3)=3, then the value of f(-3) is ______________

The function f: RvecR satisfies f(x^2)dotf^(x)=f^(prime)(x)dotf^(prime)(x^2) for all real xdot Given that f(1)=1 and f^(1)=8 , then the value of f^(prime)(1)+f^(1) is 2 b. 4 c. 6 d. 8

If graph of y=f(x) is symmetrical about the point (5,0) and f^(prime)(7)=3, then the value of f^(prime)(3) is__________

f(x+y)=f(x).f(y) for all x,yinR and f(5)=2,f'(0)=3 then f'(5) is equal to

Let f: R->R be a differentiable function with f(0)=1 and satisfying the equation f(x+y)=f(x)f^(prime)(y)+f^(prime)(x)f(y) for all x ,\ y in R . Then, the value of (log)_e(f(4)) is _______

Suppose that f(x) is differentiable invertible function f^(prime)(x)!=0a n dh^(prime)(x)=f(x)dot Given that f(1)=f^(prime)(1)=1,h(1)=0 and g(x) is inverse of f(x) . Let G(x)=x^2g(x)-x h(g(x))AAx in Rdot Which of the following is/are correct? G^(prime)(1)=2 b. G^(prime)(1)=3 c. G^(1)=2 d. G^(1)=3

If f(x)=x|x|, then prove that f^(prime)(x)=2|x|

Suppose f(x)=e^(a x)+e^(b x) , where a!=b , and that f prime prime(x)-2f^(prime)(x)-15f(x)=0 for all xdot Then the value of (|a b|)/3 is ___

Let f ''(x) be continuous at x=0 If ("lim")_(xvec0)(2f(x)-3a(f(2x)+bf(8x))/(sin^2x) exists and f(0)!=0,f^(prime)(0)!=0, then the value of (3a)/b is____

If f((x+y)/3)=(2+f(x)+f(y))/3 for all x,y f'(2)=2 then find f(x)