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[parent] derivative for parametric form (Derivation)

Instead of the usual way $ y = f(x)$ to present plane curves it is in many cases more comfortable to express both coordinates, $ x$ and $ y$, by means of a suitable auxiliary variable, the parametre. It is true e.g. for the cycloid curve.

Suppose we have the parametric form

$\displaystyle x = x(t),\quad y = y(t).$ (1)

For getting now the derivative $ \displaystyle\frac{dy}{dx}$ in a point $ P_0$ of the curve, we chose another point $ P$ of the curve. If the values of the parametre $ t$ corresponding these points are $ t_0$ and $ t$, we thus have the points $ (x(t_0),\,y(t_0))$ and $ (x(t),\,y(t))$ and the slope of the secant line through the points is the difference quotient
$\displaystyle \frac{y(t)-y(t_0)}{x(t)-x(t_0)} = \frac{\frac{y(t)-y(t_0)}{t-t_0}}{\frac{x(t)-x(t_0)}{t-t_0}}.$ (2)

Let us assume that the functions (1) are differentiable when $ t = t_0$ and that $ x'(t_0) \neq 0$. As we let $ t\to t_0$, the left side of (2) tends to the derivative $ \frac{dy}{dx}$ and the right side to the quotient $ \frac{y'(t_0)}{x'(t_0)}$. Accordingly we have the result
$\displaystyle \left(\frac{dy}{dx}\right)_{\!t=t_0} =\, \frac{y'(t_0)}{x'(t_0)}.$ (3)

Note that the formula (3) may be written
$\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}.$

Example. For the cycloid

$\displaystyle x = a(\varphi-\sin{\varphi}),\quad y = a(1-\cos{\varphi}),$
we obtain
$\displaystyle \frac{dy}{dx} = \frac{\frac{d}{d\varphi}(1-\cos\varphi)}{\frac{d}... ...rphi-\sin\varphi)} = \frac{\sin\varphi}{1-\cos\varphi} = \cot\frac{\varphi}{2}.$



"derivative for parametric form" is owned by pahio.
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See Also: goniometric formulas, curvature of Nielsen's spiral, parameter

Keywords:  parametric form

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Cross-references: quotient, side, differentiable, functions, difference quotient, secant line, slope, point, derivative, parametric form, curve, cycloid, parametre, auxiliary variable, coordinates, plane curves
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This is version 6 of derivative for parametric form, born on 2007-08-29, modified 2008-03-23.
Object id is 9904, canonical name is DerivativeForParametricForm.
Accessed 552 times total.

Classification:
AMS MSC: 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
 46G05 (Functional analysis :: Measures, integration, derivative, holomorphy :: Derivatives)
 26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
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