Consider the function \( f(x)=\begin{cases} x^2\ln|x|, & x\neq 0,\\[4pt] 0, & x=0. \end{cases} \) What is \(f'(0)\) equal to?
0
Solution
Use the definition of derivative at 0:
\[ f'(0)=\lim_{h\to 0}\frac{f(h)-f(0)}{h} =\lim_{h\to 0}\frac{h^2\ln|h|-0}{h} =\lim_{h\to 0}h\ln|h|. \]
The limit \( \lim_{h\to 0} h\ln|h| = 0\) (since \(|\ln|h||\) grows much more slowly than \(1/|h|\)).
Therefore, \( \displaystyle f'(0)=0.\)
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