Derivative of a Function Represented Parametrically

If the system of equations
MATH
where $f(t)$ and $g(t)$ are differentiable functions and MATH, defines $y$ as a continuous function of $x$ , then the derivative $\frac{dy}{dx}$ of $y$ with respect to $x$ exists and
MATH

The derivatives of higher orders are computed successively:
MATH

MATH
and so on.

Example. If $x=a(t-\sin \ t)$, $y=a(1-\cos \ t)$, then
MATH

Example. If MATH, MATH, then
MATH

Example. If $x=e^{-t}$, $y=t^{3}$, then
MATH

MATH

MATH

@