<s>[INST] Differentiation of logarithmic functions using the chain rule What's the derivative of $x^2(\ln(x^2))$?
I'm having a really hard time with logarithmic differentiation. Can someone help rationalize it for me?
 [/INST] Hints:


*

*$\bigl( f \cdot g \bigr)' = f \cdot g' + g \cdot f'$.


Using this we see that if we see that 
\begin{align*}y &= x^{2} \cdot \log(x^{2})\\
\implies y' &= x^{2} \cdot \frac{d}{dx}(\log(x^{2})) + \log(x^{2}) \cdot \frac{d}{dx} (2x) \\ &= x^{2} \cdot \frac{d}{dx} (2\cdot \log(x)) + \log(x^{2}) \times 2 \\
&=2 \cdot x^{2} \cdot \frac{1}{x} \cdot \log(x) + 4 \cdot \log(x)
\\ &= 2x \cdot \log(x) + 4 \cdot \log(x)
\end{align*}
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