COMEDK 2024 Shift 1 Paper

If sin‌y=x(cos(a+y)), then find ‌

dy

dx

when x=0
Differentiating both sides of the equation with respect to x, we get
‌cos‌y‌

dy

dx

=cos(a+y)+x(−sin‌(a+y))‌

dy

dx

‌cos‌y‌

dy

dx

+xsin‌(a+y)‌

dy

dx

=cos(a+y)
‌‌

dy

dx

(cos‌y+xsin‌(a+y))=cos(a+y)
‌

dy

dx

=‌

cos(a+y)

cos‌y+xsin‌(a+y)

When x=0, we have
‌

dy

dx

=‌

cos(a+y)

cos‌y

Since sin‌y=x‌cos(a+y), when x=0, we have sin‌y=0. This implies y=0 (since we are looking for the derivative at x=0 ).
Therefore, when x=0,
‌

dy

dx

=‌

cos(a+0)

cos‌0

=cos‌a
So the correct option is C.