Dear Rishabh

I think there is some mistake in question 1

confirm the qustion and try to do it by taking log

question2

dy/dx= (dy/dt)/(dx/dt)

dy/dt=2a

dx/dt=2at

dy/dx=1/t

d²y/dx²= d/dx(dy/dx)

use chain rule

d²y/dx²= d/dt(dy/dx)*dt/dx

=d/dt(1/t)*1/2at

=-(1/t²)*1/2at

=-1/2at³

xyd²y/dx²=at²*2at*(-1/2at³)= -a

 All the best.                                                          

AKASH GOYAL

AskiitiansExpert-IIT Delhi

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1)

y=e^{x+e^x+e^x+...........∞}

Y =e^x+y 

dy/dx = e^x+y(1+dy/dx)

dy/dx = (e^x+y/1-e^x+y )

2)

x=at² ...........1

y=2at    ..............2

x/y = t/2   .........3

put t in eq 2

y = 4ax/y

y² = 4ax

now differentiating above eq

2ydy/dx = 4a         .............1

again differentiating

2[(dy/dx)² +yd²y/dx²] = 0        

d²y/dx² = -(dy/dx)²/y          ..........2

multiply above eq by xy

(d²y/dx²)xy = -x(dy/dx)²

                      =-x(2a/y)²                    (from eq 1)

approve if u like my ans