Here is an alternative way to evaluate higher derivatives of a function f that may save time.

Question:

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.

eх sin t? dt f (x)