For a plane curve given by parametric equations 
and 
,
the parametric derivative is
 |
at points where 
.
Differentiating again gives
 |
These formulas give the slope and concavity behavior of the curve without first eliminating the parameter.
Parametric Derivative
See also
Derivative, Parametric Equations, Tangent VectorExplore with Wolfram|Alpha
References
Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, 1967.Cite this as:
Weisstein, Eric W. "Parametric Derivative." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ParametricDerivative.html