Problem 1. Suppose that f(n)(a) and g(") (a) exist. Prove the Leibniz's formula (product rule for higher order derivatives) by induction: n (S :9)() (a) = E(") s() (a) · gln-k) (a), k=0 where (a) =...

k > 1, (":") - (") - (,") m +1 т + k m k (e) in (c) and (d). Complete the induction proof by showing the inductive step using the formulas "/>
Extracted text: Problem 1. Suppose that f(n)(a) and g(") (a) exist. Prove the Leibniz's formula (product rule for higher order derivatives) by induction: n (S :9)() (a) = E(") s() (a) · gln-k) (a), k=0 where (a) = (6) f(0 ( f(4) (a):g(n-k) (п-0) (a)+ f(1)(a (a)+. . + k=0 with f(0) = f, g(0) = g. Also, n(п — 1) (п — 2)...2.1 1) (k(k – 1) · - n! (n – k)!k! ((п — к)(п — k — 1) . -2 . ... 2.1) with 0! = 1. (a) Write out and prove the base case for the induction. (b) Write out the inductive hypothesis and what you shall prove for inductive step. (c) Show that for any positive integer n, -0-- = 1. (d) Show the following equation for any positive integer m > k > 1, (":") - (") - (,") m +1 т + k m k (e) in (c) and (d). Complete the induction proof by showing the inductive step using the formulas