The second order derivative of `a sin^3 t ` w.r.t, `a cos^3 t` at `t = pi/4` is

Options

A. `(4sqrt(2))/(3a)`
B. `2`
C. `(1)/(12a)`
D. None of these

Answer

A

Explanation

Let `y=sin^(3)t` and `x=acos^(3)t`
On differentiating w.r.t. t, we get
`(dy)/(dt)=3asin^(2)cost`
and `(dx)/(dt)=3acos^(2)t(-sint)`
`therefore(dy)/(dx)=((dy)/(dt))/((dx)/(dt))=(3asin^(2)tcost)/(3acos^(2)t(-sint))`
`=-tant`
Again, differentiating w.r.t. x, we get
`(d^(2)y)/(dx^(2))=-sec^(2)t(dt)/(dx)=-(sec^(2)t)/(3acos^(2)t(-sint))`
`=(1)/(3a)((sec^(4)t)/(sint))`
`=((d^(2)y)/(dx^(2)))_(t=(pi)/(4))=(1)/(3a)*(4)/((1)/(sqrt(2)))=(4sqrt(2))/(3a)`