1

### IIT-JEE 1986

Fill in the Blanks
The derivative of $${\sec ^{ - 1}}\left( {{1 \over {2{x^2} - 1}}} \right)$$ with respect to $$\sqrt {1 - {x^2}}$$ at $$x = {1 \over 2}$$ is ...............

$$4$$
2

### IIT-JEE 1985

Fill in the Blanks
If $$f\left( x \right) = {\log _x}\left( {In\,x} \right),$$ then $$f'\left( x \right)$$ at $$x=e$$ is ................

$$1/e$$
3

### 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 ...........

Zero
4

### 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 $${{dy} \over {dx}} = ..........$$

$${{2 + 2x - 2{x^2}} \over {{{\left( {{x^2} + 1} \right)}^2}}}\sin {\left( {{{2x - 1} \over {{x^2} + 1}}} \right)^2}$$

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NEET

Class 12