If $$f\left( x \right) = {\log _x}\left( {In\,x} \right),$$ then $$f'\left( x \right)$$ at $$x=e$$ is ................
Answer
$$1/e$$
2
IIT-JEE 1985
Fill in the Blanks
If $${f_r}\left( x \right),{g_r}\left( x \right),{h_r}\left( x \right),r = 1,2,3$$ are polynomials in $$x$$ such that $${f_r}\left( a \right) = {g_r}\left( a \right) = {h_r}\left( a \right),r = 1,2,3$$
and $$F\left( x \right) = \left| {\matrix{
{{f_1}\left( x \right)} & {{f_2}\left( x \right)} & {{f_3}\left( x \right)} \cr
{{g_1}\left( x \right)} & {{g_2}\left( x \right)} & {{g_3}\left( x \right)} \cr
{{h_1}\left( x \right)} & {{h_2}\left( x \right)} & {{h_3}\left( x \right)} \cr
} } \right|$$ then $$F'\left( x \right)$$ at $$x = a$$ is ...........
Answer
Zero
3
IIT-JEE 1982
Fill in the Blanks
If $$y = f\left( {{{2x - 1} \over {{x^2} + 1}}} \right)$$ and $$f'\left( x \right) = \sin {x^2}$$, then $${{dy} \over {dx}} = ..........$$