COMEDK 2024 Shift 2 Paper

To find the value of ‌

d2x

dy2

, we start with the given relationships:
y=f(x),‌‌p=‌

dy

dx

,‌‌q=‌

d2y

dx2

First, we know from the given definitions:
p=‌

dy

dx

Taking the reciprocal of this derivative gives:
‌

dx

dy

=‌

1

p

Next, we need to differentiate ‌

dx

dy

with respect to y to find the second derivative of x with respect to y.
‌

d2x

dy2

=‌

d

dy

(‌

dx

dy

)=‌

d

dy

(‌

1

p

)
Using the chain rule, we can write:
‌

d

dy

(‌

1

p

)=‌

d

dx

(‌

1

p

)⋅‌

dx

dy

We already have ‌

dx

dy

=‌

1

p

, so we need to find ‌

d

dx

(‌

1

p

). Using the chain rule for differentiation:
‌

d

dx

(‌

1

p

)=−‌

1

p2

⋅‌

dp

dx

We need the derivative of p with respect to x, which is given by the second derivative of y with respect to x :
‌

dp

dx

=‌

d2y

dx2

=q
Substituting this into our equation:
‌

d

dx

(‌

1

p

)=−‌

q

p2

Now plugging this back into the earlier expression for ‌

d2x

dy2

:
‌

d2x

dy2

=−‌

q

p2

⋅‌

1

p

=−‌

q

p3

Therefore, the correct option is:
Option C: −‌

q

p3