# Parametric derivative

In calculus, a parametric derivative is a derivative of a dependent variable y with respect to an independent variable x that is taken when both variables depend on an independent third variable t, usually thought of as "time" (that is, when x and y are given by parametric equations in t ).

## First derivative

Let $x(t)\,$ and $y(t)\,$ be the coordinates of the points of the curve expressed as functions of a variable t:

$y=y(t),\quad x=x(t).$ The first derivative implied by these parametric equations is

${\frac {dy}{dx}}={\frac {dy/dt}{dx/dt}}={\frac {{\dot {y}}(t)}{{\dot {x}}(t)}},$ where the notation ${\dot {x}}(t)$ denotes the derivative of x with respect to t, for example. This can be derived using the chain rule for derivatives:

${\frac {dy}{dt}}={\frac {dy}{dx}}\cdot {\frac {dx}{dt}}$ and dividing both sides by ${\frac {dx}{dt}}$ to give the equation above.

In general all of these derivatives — dy / dt, dx / dt, and dy / dx — are themselves functions of t and so can be written more explicitly as, for example, ${\tfrac {dy}{dx}}(t).$ ## Second derivative

The second derivative implied by a parametric equation is given by

 ${\frac {d^{2}y}{dx^{2}}}$ $={\frac {d}{dx}}\left({\frac {dy}{dx}}\right)$ $={\frac {d}{dt}}\left({\frac {dy}{dx}}\right)\cdot {\frac {dt}{dx}}$ $={\frac {d}{dt}}\left({\frac {\dot {y}}{\dot {x}}}\right){\frac {1}{\dot {x}}}$ $={\frac {{\dot {x}}{\ddot {y}}-{\dot {y}}{\ddot {x}}}{{\dot {x}}^{3}}}$ by making use of the quotient rule for derivatives. The latter result is useful in the computation of curvature.

## Example

For example, consider the set of functions where:

$x(t)=4t^{2}\,$ and

$y(t)=3t.\,$ Differentiating both functions with respect to t leads to

${\frac {dx}{dt}}=8t$ and

${\frac {dy}{dt}}=3,$ respectively. Substituting these into the formula for the parametric derivative, we obtain

${\frac {dy}{dx}}={\frac {\dot {y}}{\dot {x}}}={\frac {3}{8t}},$ where ${\dot {x}}$ and ${\dot {y}}$ are understood to be functions of t.