Question: if x=a cos t and y=b sin t, then find d2y/de2
The correct answer is -
To find d^2y/dt^2, we need to take the second derivative of y with respect to t:
dy/dt = b cos t (using the chain rule)
d^2y/dt^2 = -b sin t
Now, to find d^2y/de^2, we can use the chain rule again:
d^2y/de^2 = d/dt (d/dt (y)) / (de/dt)^2
Since x = a cos t, we can find dt/de:
dt/de = 1 / dx/dt = 1 / (-a sin t) (using the chain rule)
(de/dt)^2 = (-a sin t)^2
Now, we can substitute the expressions for d^2y/dt^2 and (de/dt)^2:
d^2y/de^2 = d/dt (-b sin t) / (-a sin t)^2 = b cos t / (a^2 sin^2 t)
So the final answer is:
d^2y/de^2 = b cos t / (a^2 sin^2 t)
