Finding Taylor approximation for $x^4e^{-x^3}$

Question

I'm trying to find Taylor approximation for the function: $$x^4e^{-x^3}$$

I started taking the first, second, third, etc. derivatives but the expression for it seems to explode with terms. I was just wondering is there a trick for this one or do I just have to use brute force in order to discover the pattern? :)

thnx for any help

Answer 1

Notice $$\exp(x) = \sum \frac{x^n}{n!} \implies \exp(-x^3) = \sum\frac{(-1)^nx^{3n}}{n!} \implies x^4 \exp(-x^3) = \sum\frac{(-1)^nx^{3n+4}}{n!} $$