COMEDK2025MathematicsIndefinite IntegrationActual
If x=2 a ⁻¹ ( y 2 a )- 2 a y-y^2 then d y d x =
Options
- Ay 2 a-y
- B2 a-y y
- C1 2 ( 2 a-y y )
- D2 a-y y
Correct answer
D. 2 a-y y
Step-by-step solution
Given x = 2a ⁻¹ ( y 2a ) - 2ay - y^2 . Let y = 2a ^2 . Then y 2a = , so = ⁻¹ ( y 2a ) . Also, 2ay - y^2 = 2a(2a ^2 ) - (2a ^2 )^2 = 4a^2 ^2 (1 - ^2 ) = 4a^2 ^2 ^2 = 2a = a (2 ) . Substituting these into the expression for x : x = 2a - a (2 ) . Differentiating x with respect to : dx d = 2a - 2a (2 ) = 2a(1 - (2 )) = 2a(2 ^2 ) = 4a ^2 . Differentiating y = 2a ^2 with respect to : dy d = 2a(2 ) = 4a . Using the chain rule dy dx = dy/d dx/d : dy dx = 4a 4a ^2 = = . Since ^2 = y 2a , we have ^2 = 1 - y 2a = 2a - y 2a .