If `x=f(t)` and `y=g(t)` , then write the value of `(d^2y)/(dx^2)` .

Options

A. `(f'(t).g^(n) (t)- g'(t).f''(t))/([f'(t)]^(3))`
B. `(f'(t).g''(t)-g'(t).f''(t))/([f'(t)]^(2))`
C. `(g'(t).f''(t)-f'(t).g''(t))/([f'(t)]^(3))`
D. `(g'(t).f''(t)+f'(t).g''(t))/([f'(t)]^(3))`

Answer

A

Explanation

Given , x = f(t) and `y =g(t)`
On differentiating both sides w.r.t. 'f', we get
`(dx)/(dt) = f'(t)` and `(dy)/(dt) =g'(t)`
We know that, `(dy)/(dx) = ((dy)/(dt))/((dx)/(dt))`
`rArr (dy)/(dx) = (g'(t))/(f'(t))`
Again, differentiating both sids w.r.t. 'x', we get
`(d^(2)y)/(dx^(2)) = (f'(f).g^(n)(t)-g^'(t).f^(n)(t))/((f'(t))^(2)) .(dt)/(dx)`
`= (f'(t).g^('')(t)-g'(t).f''(t))/((f'(t))^(2)).(1)/(f'(t))`
`= (f'(t).g''(t)-g'(t).f''(t))/((f'(t))^(3))`