Evaluate \(f^{\left (15 \right)}(0) \)

[Tolaso J Kos]

Let \( f(x)=\sin \left (x^3 \right) \). Evaluate \(f^{(15)}(0) \).

[achilleas]

It's known that

\(\sin x=\sum_{n=0}^{\infty}(-1)^n\dfrac{x^{2n+1}}{(2n+1)!}=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots\).

for all \(x\in \mathbb{R}\). Hence

\(\sin (x^3)=\sum_{n=0}^{\infty}(-1)^n\dfrac{x^{3(2n+1)}}{(2n+1)!}=x^3-\dfrac{x^9}{3!}+\dfrac{x^{15}}{5!}-\cdots\).

for all \(x\in \mathbb{R}\). Since

\(f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(0)}{n!}x^n\)

for all \(x\), comparing the coefficients of \(x^{15}\) we see that

\(\dfrac{f^{(15)}(0)}{15!}=\dfrac{1}{5!}\).

Therefore,

\(f^{(15)}(0)=\dfrac{15!}{5!}\)