<s>[INST] 10th derivative of a function I want to find $f^{(10)}(0)$ where $f(x)=\ln(2+x^2)$.  
I know that it can be done "by hand", but I believe there is a smarter way.  
I think I should use Taylor series and the fact that $f^{(n)}(0)=a_n*n!$ , but I'm not sure how.  
 [/INST] We have
$$
\frac{2x}{x^2+2} = \frac{x}{1+\frac{x^2}{2}} = x\bigg(1-\frac{x^2}{2} + \frac{x^4}{4} - \frac{x^6}{8} + \frac{x^8}{16}+\cdots \bigg).
$$
Then integrating we get
$$
\log(x^2+2)-\log2 = \int_0^x \frac{2t}{t^2+2}\; dt = \frac{x^2}{2}-\frac{x^4}{2\cdot 4} + \frac{x^6}{4\cdot 6} - \frac{x^8}{8\cdot 8} + \frac{x^{10}}{10\cdot 16}+\ldots
$$
Answer: $22680$ :)
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