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

Differentiate sqrtsinx from first principles.

Let a, b, c be positive real numbers such that b^2- 4ac > 0 and let alpha_1 = c . Prove by induction that alpha_(n+1)= (a alpha_n^2)/(b^2 -2a(alpha_1+alpha_2+...+alpha_n)) is well defined and alpha_(n+1) ltalpha_n/2 for all n = 1, 2, ..... (Here, ‘well defined’ means that the denominator in the expression for alpha_(n+1) is not zero).

If f(x) = alpha x^n , prove that alpha= (f'(1))/n .

Using mathematical induction prove that d(x^n)/dx = nx^(n-1) for all positive integers n.

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

A_(alpha )= [(cos alpha, sin alpha),(- sin alpha, cos alpha)] then prove that (A_(alpha)^(n) = A(n alpha) " where " n in N

Prove the following by using the principle of mathematical induction for all n in N :- n(n +1) (n + 5) is a multiple of 3.

Prove the following by using the principle of mathematical induction for all n in N :- 10^(2n-1) + 1 is divisible by 11.