Q.303. To find the second derivative.

Question 303.
Find (d^2 y)/dx^2 (second derivative) if  x=e^t * cost and y=e^t * sint.

Answer 303.
x = e^t * cost
=> dx/dt = e^t * cost - e^t * sint = x - y

y = e^t * sint
=> dy/dt = e^t * sint + e^t * cost = x + y
=> dy/dx = (dy/dt) / (dx/dt) = (x + y) / (x - y)

=> d^2y/dx^2
= [(x - y) * (1 + dy/dx) - (x +y) * (1 - dy/dx)] / (x - y)^2
= [(x - y) * (1 + (x+y)/(x-y)) - (x + y) * (1 - (x+y)/(x-y))] / (x - y)^2
= [2x + 2y(x+y)/(x-y)] / (x - y)^2
= (2x^2 - 2xy + 2xy + 2y^2) / (x - y)^3
= [2(x^2 + y^2)] / (x - y)^3.

Link to YA!