Correct Answer - Option 3 : not defined
Concept:
Suppose that we have two functions f(x) and g(x) and they are both differentiable.
- Chain Rule: \(\frac{{\rm{d}}}{{{\rm{dx}}}}\left[ {{\rm{f}}\left( {{\rm{g}}\left( {\rm{x}} \right)} \right)} \right] = {\rm{\;f'}}\left( {{\rm{g}}\left( {\rm{x}} \right)} \right){\rm{g'}}\left( {\rm{x}} \right)\)
- Product Rule: \(\frac{{\rm{d}}}{{{\rm{dx}}}}\left[ {{\rm{f}}\left( {\rm{x}} \right){\rm{\;g}}\left( {\rm{x}} \right)} \right] = {\rm{\;f'}}\left( {\rm{x}} \right){\rm{\;g}}\left( {\rm{x}} \right) + {\rm{f}}\left( {\rm{x}} \right){\rm{\;g'}}\left( {\rm{x}} \right)\)
Calculation:
Let f(x) = \(\sqrt{\rm x^{2} - 4}\)
On differentiating , we get
f'(x) = \(\frac{1}{2\sqrt{x^{2}-4}}\) × 2x
= \(\frac{x}{\sqrt{x^{2}-4}}\)
Putting x = 2 , we get
f'(x) = \(\frac{2}{0}\) = not defined