Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Explain why f(k)(a) = k! multiplied by the coefficient of (x ? a)k.Use this idea to evaluate f(3)(0) and f(4)(0) for the following functions. Use known series and do not evaluate derivatives.
Solution 68EStep 1:In this question we will use the known taylor series of to evaluate the taylor series of And then integrating the taylor series of to get the desired function The taylor series of a function centered at zero is given by :Using the above series, we find that Coefficient of Hence we can write that Step 2:The taylor series for is :Substituting in place of , we get