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derivative for parametric form
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(Derivation)
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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
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(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
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(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
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(3) |
Note that the formula (3) may be written $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}.$$
Example. For the cycloid $$x = a(\varphi-\sin{\varphi}),\quad y = a(1-\cos{\varphi}),$$ we obtain $$\frac{dy}{dx} = \frac{\frac{d}{d\varphi}(1-\cos\varphi)}{\frac{d}{d\varphi}(\varphi-\sin\varphi)} = \frac{\sin\varphi}{1-\cos\varphi} = \cot\frac{\varphi}{2}.$$
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"derivative for parametric form" is owned by pahio.
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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
There is 1 reference to this entry.
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 1494 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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Pending Errata and Addenda
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