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प्रश्न
If x = f(t) and y = g (t) are differentiable functions of t, then `(d^2y)/(dx^2)` is ______.
विकल्प
`(f^'(t).g^('')(t) - g^'(t).f^('')(t))/[f^'(t)]^3`
`(f^'(t).g^('')(t) - g^'(t).f^('')(t))/[f^'(t)]^2`
`(g^'(t).f^('')(t) - f^'(t).g^('')(t))/[f^'(t)]^3`
`(g^'(t).f^('')(t) + f^'(t).g^('')(t))/[f^'(t)]^3`
उत्तर
If x = f(t) and y = g (t) are differentiable functions of t, then `(d^2y)/(dx^2)` is `underlinebb((f^'(t).g^('')(t) - g^'(t).f^('')(t))/[f^'(t)]^3`.
Explanation:
We have x = f'(t) and y = g(t)
After differentiating on both sides w.r.t 't', we get
`dx/dt` = f'(t) and `dy/dt` = g'(t)
As, `dy/dx = (dy/dt)/(dx/dt)`
`\implies dy/dx = g^'(t)/(f^'(t)`
Now, differentiating again both sides w.r.t. 'x', we get
`(d^2y)/(dx^2) = (f^'(t).g^('')(t) - g^'(t).f^('')(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`
