UPSC NDA Math Model Paper 7

This is the case of both Composite Functions and Parametric Functions. Both x and y are given as composite functions in terms of the parameter θ.
Let us first find out

dx

dθ

and

dy

dθ

by using the Chain Rule for Composite Functions.
x=asec2θ
⇒

dx

dθ

=a

d

dθ

(sec2θ) =a×

d

d(sec‌θ)

(sec2θ)×

d

dθ

‌sec‌θ
⇒

dx

dθ

=a×2‌sec‌θ×tan‌θ‌sec‌θ =2‌a‌tan‌θ‌sec2‌θ
y=atan2θ
⇒

dy

dθ

=a

d

dθ

(tan2θ)=a×

d

d(tan‌θ)

(tan2θ) ×

d

dθ

‌tan‌θ
⇒

dy

dθ

=a×2‌tan‌θ×sec2θ =2‌a‌tan‌θ‌sec2‌θ
∴

dy

dx

=

dy

dθ

×

dθ

dx

⇒

dy

dx

=2‌a‌tan‌θ‌sec2‌θ‌ ×

1

2‌a‌tan‌θ‌sec2‌θ

=1
And finally,

d2y

dx2

=

d

dx

(

dy

dx

)=

d

dx

(1)=0