Tell me if my solution is correct for this one:
Q: Given f(x)=cos(x^3), find $f^{(15)}(0).$
S: The Maclaurin series of f(x) is given by
$$\sum_{n=0}^{\infty} (-1)^{n+1}\frac{(x^3)^{2n}}{(2n)!}$$.
The coefficient of x^15 is 0; therefore, $\frac{f^{(15)}(0)}{15!}=0$ and consequently, $f^{(15)}(0)=0$.

Question

Tell me if my solution is correct for this one:
Q: Given f(x)=cos(x^3), find $f^{(15)}(0).$
S: The Maclaurin series of f(x) is given by
$$\sum_{n=0}^{\infty} (-1)^{n+1}\frac{(x^3)^{2n}}{(2n)!}$$.
The coefficient of x^15 is 0; therefore, $\frac{f^{(15)}(0)}{15!}=0$ and consequently, $f^{(15)}(0)=0$.

anonymous · Answer

seems legit to me, i would have just said that all the odd derivatives are in the form of
$$C*x^n*\sin(x^3)$$
but that works too