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Find the derivative of \(\sqrt{\rm x^{2} - 4}\)   at x = 2

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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

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