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Question
Let f(x) be a polynomial. Then, the second order derivative of f(ex) is
Options
f'' (ex) e2x + f'(ex) ex
f'' (ex) ex + f' (ex)
f'' (ex) e2x + f'' (ex) ex
f'' (ex)
Solution
(a) f''(ex)e2x + f'(ex)ex
Since f(x) is a polynomial,
\[\therefore f^{'} \left( e^x \right) = f^{'} \left( e^x \right) e^x \]
\[ \Rightarrow f^{''} \left( e^x \right) = f^{''} \left( e^x \right) ( e^x )^2 + f^{'} \left( e^x \right) e^x \]
\[ = f^{''} \left( e^x \right) e^{2x} + f^{'} \left( e^x \right) e^x\]
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