Need help answering proof question about differentiating on an interval

Question

I am unsure about how to approach this question about differentiability and would appreciate some help on how to prove this.

Suppose that g is differentiable on an interval I. Show that for all n in N, g^n  is differentiable on I.  by definition, g^n(x) = (g(x))^n

Answer 1

By induction..........

G is differentiable on I is given, so dG exists on I.

derivative of G^2 = 2*G dG. Since dG exists on I, so does the derivative of G^2.

suppose the derivative of G^k exists on I.

The derivative of y = G^(k+1) is (k+1) * G^k dG, which exists on I since

G^k and dG exists on I.

[end of proof]