Mathematics

Problem

Let

x=t^{2}

and

y=t^{3}

, where

t\in\mathbb R

. Show that the second derivative obtained by differentiating

\dfrac{dy}{dx}

with respect to

x

satisfies

\frac{d^{2}y}{dx^{2}}=\frac{f'(t)\,g''(t)-g'(t)\,f''(t)}{[f'(t)]^{3}},

where

f(t)=t^{2}

and

g(t)=t^{3}

, and hence find

\dfrac{d^{2}y}{dx^{2}}

in terms of

t