a. \(f(x) = ax + 3\), cho x0 một số gia Δx, ta có:
\(\eqalign{ & \Delta y = f\left( {{x_0} + \Delta x} \right) - f\left( {{x_0}} \right) \cr & = a\left( {{x_0} + \Delta x} \right) + 3 - \left( {a{x_0} + 3} \right) = a\Delta x \cr & \Rightarrow {{\Delta y} \over {\Delta x}} = a \Rightarrow f'\left( {{x_0}} \right) = \mathop {\lim }\limits_{\Delta x \to 0} {{\Delta y} \over {\Delta x}} = a \cr} \)
b.
\(\eqalign{ & f\left( x \right) = {1 \over 2}a{x^2},\Delta y = f\left( {{x_0} + \Delta x} \right) - f\left( {{x_0}} \right) \cr & = {1 \over 2}a{\left( {{x_0} + \Delta x} \right)^2} - {1 \over 2}ax_0^2 \cr & = {1 \over 2}a\Delta x\left( {2{x_0} + \Delta x} \right) \cr & \Rightarrow f'\left( {{x_0}} \right) = \mathop {\lim }\limits_{\Delta x \to 0} {{\Delta y} \over {\Delta x}} \cr & = \mathop {\lim }\limits_{\Delta x \to 0} {1 \over 2}a\left( {2{x_0} + \Delta x} \right) = a{x_0} \cr} \)
