A Power Series Is Better Than The Quotient Rule

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$ $\frac{1}{1+x^3}=1-x^3+x^6-x^9+x^{12}-...$ The previous equation is obtained by viewing $\frac{1}{1+x^3}$ as a geometric series with first term $1$ and a common ratio of $-x^3$ . This is of course if $|x^3|<1$ . Now, $f(x)=(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...)(1-x^3+x^6-x^9+x^{12}-...)$ We are only interested in combining those terms that will result to a degree of 7. Those that are of degree less than 7 will become zero by getting their seventh derivatives and those that are higher than 7 will become zero too , because they still contain $x$ upon getting the seventh derivative and substituting $x=0.$ The seventh degree terms are: $\frac{x^7}{7!}-\frac{x^7}{4!}+\frac{x^7}{1!}$ and the seventh derivative at $x=0$ is $\frac{7!}{7!}-\frac{7!}{4!}+\frac{7!}{1!}=4831$