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**MathCrusader** I am to determine $\displaystyle f^{(3)}(0)$ by using Taylor expansion about $\displaystyle x= 0$ for

$\displaystyle f(x) = \sin^3 (\ln (1+x)) \, .$

In our toolkit, we have

$\displaystyle \ln (1+x) = x - \frac {x^2}2 + \frac {x^3}3 - \frac {x^4}4 + \text{O}(x^5)\, , \ \ x \rightarrow 0 \, .$

$\displaystyle \sin (x) = x - \frac {x^3}{3!} + \frac {x^5}{5!} - \frac {x^7}{7!} + \text{O}(x^9)\, , \ \ x \rightarrow 0 \, .$

Now, plugging in the polynomial for $\displaystyle \ln (1+x)$ into the polynomial for $\displaystyle \sin (x)$ and then cubing(!!!) it all sounds not like something fun at all to do, but I have done similar problems for and that is really the best(?) way to go.

Do you guys have any tips for how I can make the polynomial multiplication easier and quickly obtain the sought coefficient, instead of having to really multiply out too many terms and add upp their coefficients etc.