Show that `f(x)=x^2` is differentiable at `x=1` and find `f^(prime)(1)` .

Show that f(x)=x^(1//3) is not differentiable at x=0 .

Show that f(x)=(x)^(1/3) is not differentiable at x=0.

A function f: R->R satisfies that equation f(x+y)=f(x)f(y) for all x ,\ y in R , f(x)!=0 . Suppose that the function f(x) is differentiable at x=0 and f^(prime)(0)=2 . Prove that f^(prime)(x)=2\ f(x) .

A function f: R->R satisfies that equation f(x+y)=f(x)f(y) for all x ,\ y in R , f(x)!=0 . Suppose that the function f(x) is differentiable at x=0 and f^(prime)(0)=2 . Prove that f^(prime)(x)=2\ f(x) .

Let f: Rvec satisfying |f(x)|lt=x^2AAx in R be differentiable at x=0. The find f^(prime)(0)dot

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

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^(prime)(1)=2 d. G^(prime)(1)=3

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

Show that f(x)={12 x-13 ,\ \ \ if\ xlt=3 \ \ \ \ \ 2x^2+5,\ \ \ \ \ if\ x >3 is differentiable at x=3 . Also, find f^(prime)(3) .

Show that f(x)={12 x-13 , ifxlt=3 , 2x^2+5 ifx >3 is differentiable at x=3 . Also, find f^(prime)(3)dot