b) In some situations, we would like to differentiate a function more than once. We callthis higher-order differentiation, and we can define it in the following way:f(0(x) = f (x)fla+1) (x) = (f") (x)Calculate fa Chowyour working,ii) Now let f(x) be a polynomial of order n, i.e. one whose highest non-zero coeffi-cient is c,. For example, the polynomial above is of order 5, since it has highestnon-zero coefficient c5 = 2. Show that f("+1)(x) = 0iii) Now let ) = si Caleulateour workingiv) Fix some a e R. With f(x) = sin(x) as before, we define a new function g: N→Rsuch that g(n) = f(n)(a). Write an alternative definition of g(n) that does notinvolve differentiation.c) Below is an alternative but equivalent definition of the definite integral of f : R → Rbetween a and b, given x = a+ k() for 0 y implies f(x) > f (y).Show that, if f is monotone increasing, then, for any a,be R, a < b , and n e NIn(f, a,b)< Jn(f , a, b)v) Without formally proving it, explain why:lim J„(f , a, b) =f (x)dx = lim I,(f,a,b)100

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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: f(®(x) = f(x) flne1)(x) = (f\")'(x) . Caleulate f Chowyour working ii) Now let f(x) be a polynomial of order n, i.e. one whose highest non-zero coeffi- cient is c,. For example, the polynomial above is of order 5, since it has highest non-zero coefficient c5 2. Show that f+1)(x)= 0 im) Now let )=si Caleulate iv) Fix some a e 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.