<s>[INST] How to find the second derivative of $f(x)=\frac{x^2}{10+x}$ How do you find the second derivative of $f(x)=\frac{x^2}{10+x}$?
I get $f'(x)={20x+x^2}/{(10+x)^2}$
Then get stuck here:
$(10+x)^2 (20+2x) - (20x+x^2) \frac {2(10+x)}{((10+x)^2)^2}$
 [/INST] First make the Euclidean division:
$$x^2=(x-10)(x+10)+100, \enspace\text{whence}\quad\frac{x^2}{x+10}=x-10+\frac{100}{x+10}$$
Thus
\begin{align}
f'(x)&=1-\frac{100}{(x+10)^2},\\
f''(x)&=\frac{200}{(x+10)^3}.
\end{align}
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