Differentiation

b) In some situations, we would like to differentiate a function more than once. We call this higher-order differentiation, and we can define it in the following way: flo) ( x ) = f (x) f ( # + 1) ( x ) = (f ( ) )' ( x ) i) Let f(x) = 2x5 + 3x4 +4. Calculate f(3)(x) . Show your working. ii) Now let f(x) be a polynomial of order n , i.e. one whose highest non-zero coeffi- cient is cy . For example, the polynomial above is of order 5, since it has highest non-zero coefficient cs = 2. Show that f(#+1)(x) = 0 Problem Sheet 2 Mathematics for Al iii) Now let f(x) = sinx. Calculate f(5)(x) . Show your working. iv) Fix some ac R. With f(x) = sin(x) as before, we define a new function g: N - R such that g(n) = f(")(a). Write an alternative definition of g(n) that does not involve differentiation