Xét hai hàm số: \[\left( {\rm{I}} \right){\rm{: f}}\left( {\rm{x}} \right){\rm{ = }}\left| {\rm{x}} \right|{\rm{x,}}\,\,\left( {{\rm{II}}} \right){\rm{: g}}\left( {\rm{x}} \right){\rm{ = }}\sqrt {\rm{x}} \] . Hàm số có đạo hàm tại x = 0  là:

\(\left\{ {\begin{array}{*{20}{c}}{\mathop {\lim }\limits_{x \to {0^ + }} \frac{{f\left( x \right) - f\left( 0 \right)}}{{x - 0}} = \mathop {\lim }\limits_{x \to {0^ + }} \frac{{{x^2}}}{x} = \mathop {\lim }\limits_{x \to {0^ + }} x = 0}\\{\mathop {\lim }\limits_{x \to {0^ - }} \frac{{f\left( x \right) - f\left( 0 \right)}}{{x - 0}} = \mathop {\lim }\limits_{x \to {0^ - }} \frac{{ - {x^2}}}{x} = \mathop {\lim }\limits_{x \to {0^ - }} \left( { - x} \right) = 0}\end{array}} \right.\)

\[ \Rightarrow \mathop {\lim }\limits_{{\rm{x}} \to {0^ + }} \frac{{{\rm{f}}\left( {\rm{x}} \right) - {\rm{f}}\left( {\rm{0}} \right)}}{{{\rm{x}} - 0}} = \mathop {\lim }\limits_{{\rm{x}} \to {0^ - }} \frac{{{\rm{f}}\left( {\rm{x}} \right) - {\rm{f}}\left( {\rm{0}} \right)}}{{{\rm{x}} - 0}} = 0\]

⇒ Hàm số y = f(x) có đạo hàm tại x = 0.

\[\mathop {\lim }\limits_{{\rm{x}} \to {0^ + }} \frac{{{\rm{g}}\left( {\rm{x}} \right) - {\rm{g}}\left( {\rm{0}} \right)}}{{{\rm{x}} - 0}} = \mathop {\lim }\limits_{{\rm{x}} \to {0^ + }} \frac{{\sqrt {\rm{x}} }}{{\rm{x}}} = \mathop {\lim }\limits_{{\rm{x}} \to {0^ + }} \frac{1}{{\sqrt {\rm{x}} }} = + \infty \]

⇒ Hàm số y = g(x) không có đạo hàm tại x = 0.

Đáp án cần chọn là: A