Taylor series problem (non-direct differentiation?)

[Loopas]

I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?

 

[pasmith]

I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?

Multiply the Taylor series of $\sin(7x)$ by $x^3$.

 

[Loopas]

Thank you... wouldve never guessed it was so simple.