Q.215. Differentiation

Question 215.
If x = 2cos t - cos2t  &  y = 2sin t - sin2t,  then find d^2y/dx^2.

Answer 215.
x = 2cost - cos2t
=> dx/dt
= - 2sint + 2sin2t
= 2(sin2t - sint)
= 4sin(3t/2) cos(t/2) ... (1)

y = 2sint - sin2t
=> dy/dt
= 2cost - 2cos2t
= 2(cost - cos2t)
= 4sin(3t/2) sin(t/2) ... (2)

From (1) and (2),
dy/dx
= (dy/dt) / (dx/dt)
= tan(t/2)

d^2y/dx^2
= d/dx(dy/dx)
= d/dt(dy/dx) * dt/dx
= d/dt[tan(t/2)] * 1 / (dx/dt)
= (1/2) sec^2 (t/2) * 1 / [4sin(3t/2) cos(t/2)
= (1/8) sec^3 (t/2) cosec(3t/2).

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