Using mathematical induction prove that for all positive integers n.

To prove : P(n) : = nx^{n – 1} for all positive integers n

For n = 1,

LHS = = 1

RHS = 1 × x^{1 – 1} = 1

So, LHS = RHS

∴ P(1) is true.

∴ P(n) is true for n = 1

Let P(k) be true for some positive integer k.

i.e. P(k) =

Now, to prove that P(k + 1) is also true

RHS = (k + 1)x^{(k + 1) – 1}

LHS =

∴ LHS = RHS

Thus, P(k + 1) is true whenever P(k) is true.

Therefore, by the principle of mathematical induction, the statement P(n) is true for every positive integer n.

Hence, proved.