Let $$f\left( {x + y} \right) = f\left( x \right) + f\left( y \right)$$     and $$f\left( x \right) = {x^2}g\left( x \right)$$    for all $$x,\,y\, \in \,R,$$   where $$g\left( x \right)$$  is continuous function. Then $$f'\left( x \right)$$  is equal to :

Solution :
$$\eqalign{ & {\text{We have }}f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - \left( x \right)}}{h} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( x \right) + f\left( h \right) - f\left( x \right)}}{h} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( h \right)}}{h} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{{h^2}g\left( h \right)}}{h} \cr & = 0.g\left( 0 \right) \cr & = 0 \cr & \left[ {\because \,g{\text{ is continuous therefore }}\mathop {\lim }\limits_{h \to 0} g\left( h \right) = g\left( 0 \right)} \right] \cr} $$