Find d^2y/dx^2?

Question

If x=sin t and y=cos^2 (t), find d^2y/dx^2 at t=pi.

 

Can you please find d/dt and dy/dx? That's the only thing I need to solve the problem since I know dx/dt. 

Answer 1

x=sin t and y=cos^2 (t), find d^2y/dx^2

 

dx = cos t dt, dy = -2 cos t sin t dt

 

dy/dx = -2 cost t sin t /cos t = - 2 sin t = - 2 x

 

d^2y/dx^2 = - 2

Answer 2

X = sin t    Y = Cos^2t    find d²y/ d²X

 

dx/ dt = 1/ cos t   t = sin^-1X    dt/dx=1/ cost

 

dy/dt = - 2 Cos t sin t

 

dy/ dx = dy/dt . dt/dx

          = -2cost sint/cost = -2 sint= -2x

 

d²Y/ dX² = d/dx( d/dx) = d/dx( -2X) = -2