Mathematics Problem Of the Week

Fall 2001

POW #10

Products Rule

Theorem: If f,g,h, and k are differentiable functions such that

fg=hk and f'g'=h'k'

then either f^(m)g⁽ⁿ⁾=h^(m)k⁽ⁿ⁾ for all m,n

or f^(m)g⁽ⁿ⁾=h⁽ⁿ⁾k^(m) for all m,n

where f⁽ⁿ⁾ denotes the n^th derivative of f.

(a) Show by means of an example that we need the "or" condition, i.e. find functions f,g,h, and k that satisfy the hypotheses but have f^(m)g⁽ⁿ⁾¹h^(m)k⁽ⁿ⁾ for some m and n.

(b) Prove the theorem. (Hint: Think about what's true about functions that satisfy the hypotheses.)

Due Friday, December 7^th at Noon.