If a curve is represented parametrically by the equation `x=f(t) and y=g(t)" then prove that "(d^(2)y)/(dx^(2))=-[(g'(t))/(f'(t))]^(3)((d^(2)x)/(dy^(2)))`

Explanation

To prove the given statement, we will start with the parametric equations \( x = f(t) \) and \( y = g(t) \). We need to find the second derivative \( \frac{d^2y}{dx^2} \) in terms of \( g'(t) \) and \( f'(t) \).

### Step 1: Find the first derivative \( \frac{dy}{dx} \)

Using the chain rule, we can express the derivative \( \frac{dy}{dx} \) as follows:

\[
\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}
...