a. Đặt \(f(x)=y = a{x^2}\)
Với \(x_0\in\mathbb R\) ta có:
\(\eqalign{ & f'\left( {{x_0}} \right) = \mathop {\lim }\limits_{\Delta x \to 0} {{f\left( {{x_0} + \Delta x} \right) - f\left( {{x_0}} \right)} \over {\Delta x}} \cr & = \mathop {\lim }\limits_{\Delta x \to 0} {{a{{\left( {{x_0} + \Delta x} \right)}^2} - ax_0^2} \over {\Delta x}} \cr & = \mathop {\lim }\limits_{\Delta x \to 0} a\left( {2{x_0} + \Delta x} \right) = 2a{x_0} \cr} \)
b. Đặt \(f(x)=y = {x^3} + 2\)
Với \(x_0\in\mathbb R\) ta có:
\(\eqalign{ & f'\left( {{x_0}} \right) = \mathop {\lim }\limits_{\Delta x \to 0} {{f\left( {{x_0} + \Delta x} \right) - f\left( {{x_0}} \right)} \over {\Delta x}} \cr & = \mathop {\lim }\limits_{\Delta x \to 0} {{{{\left( {{x_0} + \Delta x} \right)}^3} + 2 - x_0^3 - 2} \over {\Delta x}} \cr & = \mathop {\lim }\limits_{\Delta x \to 0} \left[ {{{\left( {{x_0} + \Delta x} \right)}^2} + \left( {{x_0} + \Delta x} \right){x_0} + x_0^2} \right] = 3x_0^2 \cr} \)
