UPSC NDA Math Model Paper 8

Given, x=k(θ+sin‌θ)
⇒

dx

dθ

=k(1+cos‌θ)
y=k(1+cos‌θ)
⇒

dy

dθ

=k(−sin‌θ)
By chain rule, we have

dy

dx

=

dy dθ

dx dθ

⇒

dy

dx

=

−k‌sin‌θ

k(1+cos‌θ)

⇒

dy

dx

=

−sin‌θ

(1+cos‌θ)

Put θ = π/2

dy

dx

=

−sin‌ π 2

(1+cos‌ π 2 )

⇒

dy

dx

=−1
Hence, if x = k(θ + sin θ) and y = k (1 + cos θ), then the derivative of y with respect to x at θ = π/2 is -1.