QJEEM21Ma5955: JEE Main 2021 Mathematics - Class 12 JEEM21Ma5955 Solution: If y = y(x) is an implicit function of x such that log e (x +…

Question

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If y = y(x) is an implicit function of x such that log_e(x + y) = 4xy, then

{{{d^2}y} \over {d{x^2}}}

at x = 0 is equal to ___________.

Accepted Answer

ln(x + y) = 4xy (At x = 0, y = 1)

x + y = e^4xy

\Rightarrow 1 + {{dy} \over {dx}} = {e^{4xy}}\left( {4x{{dy} \over {dx}} + 4y} \right)

At x = 0

{{dy} \over {dx}} = 3

{{{d^2}y} \over {d{x^2}}} = {e^{4xy}}{\left( {4x{{dy} \over {dx}} + 4y} \right)^2} + {e^{4xy}}\left( {4x{{{d^2}y} \over {d{x^2}}} + 4y} \right)

At x = 0,

{{{d^2}y} \over {d{x^2}}} = {e^0}{(4)^2} + {e^0}(24)

\Rightarrow {{{d^2}y} \over {d{x^2}}} = 40

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