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In calculus, a parametric derivative is a derivative of a dependent variable with respect to another dependent variable that is taken when both variables depend on an independent third variable, usually thought of as "time" (that is, when the dependent variables are x and y and are given by parametric equations in t).
First derivative edit
Let x(t) and y(t) be the coordinates of the points of the curve expressed as functions of a variable t:
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, .
Second derivative edit
The second derivative implied by a parametric equation is given by
Example edit
For example, consider the set of functions where:
See also edit
References edit
- Derivative for parametric form at PlanetMath.
- Harris, John W. & Stöcker, Horst (1998). "12.2.12 Differentiation of functions in parametric representation". Handbook of Mathematics and Computational Science. Springer Science & Business Media. pp. 495–497. ISBN 0387947469.
- Briggs, William L; Cochran, Lyle; Gilett, Bernard; Schulz, Eric. "11 Parametric and Polar Curves". Calculus for Scientists and Engineers – Early Transcendentals. Pearson. p. 734.