Question

The p^th derivative of a q^th degree monic polynomial, where p, q are positive integers and 2p^4 + 3pq^^3⁄2 = 3q^^3⁄2 + 2qp^3 is given by?

(a) Cannot be generally determined

(b) (q – 1)!

(c) (q)!

(d) (q – 1)! * p^q

I had been asked this question in an international level competition.

Question is taken from The nth Derivative of Some Elementary Functions in division Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (c) (q)!

The best I can explain: First consider the equation

2p^4 + 3pq^^3⁄2 = 3q^^3⁄2 + 2qp^3

After simplification, we get

(2p^3 + 3q^^1⁄2) (p – q) = 0

This gives us two possibilities

2p^3 = – 3q^^1⁄2

OR

p = q

The first possibility can’t be true as we are dealing with positive integers

Hence, we get

p = q

Thus the p^th derivative of any monic polynomial of degree p(p = q) is

p! =q!.