From the first principles, prove that `d/(dx)(ax^n)=anx^(n-1)`.

Prove from first principles, that d/(dx)(alpha x^n)=alpha nx^(n-1) .

Prove that n! (n+2)=n!+(n+1)! .

Prove the following by the principle of mathematical induction: \ x^(2n-1)+y^(2n-1) is divisible by x+y

Prove the following by principle of mathematical induction : (a) If A=[{:(11,-25),(4,-9):}] , then A^(n)=[{:(1+10n,-25n),(4n,1-10n):}] where n is a positive integer .