If f(x) = sin(x^3) find f^(15)(0)

[Calcotron]

Short and sweet, If f(x) = sin(x^3) find f^(15)(0) by which I mean find the 15th derivative of f. Do I have to write out all the derivatives and look for a pattern somewhere or is there an easier way?

 

[gel]

Do you know the power series expansion for sin? If so, you can read off the derivatives from the Taylor series expansion
$$f(x) = \sum_{n=0}^\infty\frac{x^n}{n!}f^{(n)}(0).$$

 

[Calcotron]

Ok, so from the maclaurin series with x^3 substituted I get f(x) = x^3 - (x^9/ 3!) + (x^15/ 5!) and just from this I can see the 15th derivative of the function is going to be 15!/ 5!. Is that all correctÉ

 

[gel]

yes!