# derivative for parametric form

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

$x=x(t),y=y(t).$ | (1) |

For getting now the derivative^{} $\frac{dy}{dx}$ 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

$\frac{y(t)-y({t}_{0})}{x(t)-x({t}_{0})}}={\displaystyle \frac{\frac{y(t)-y({t}_{0})}{t-{t}_{0}}}{\frac{x(t)-x({t}_{0})}{t-{t}_{0}}}}.$ | (2) |

Let us assume that the functions (1) are differentiable^{} when $t={t}_{0}$ and that ${x}^{\prime}({t}_{0})\ne 0$. As we let $t\to {t}_{0}$, the left side of (2) tends to the derivative $\frac{dy}{dx}$ and the side to the quotient $\frac{{y}^{\prime}({t}_{0})}{{x}^{\prime}({t}_{0})}$. Accordingly we have the result

${\left({\displaystyle \frac{dy}{dx}}\right)}_{t={t}_{0}}={\displaystyle \frac{{y}^{\prime}({t}_{0})}{{x}^{\prime}({t}_{0})}}.$ | (3) |

Note that the (3) may be written

$$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}.$$ |

Example. For the cycloid

$$x=a(\phi -\mathrm{sin}\phi ),y=a(1-\mathrm{cos}\phi ),$$ |

we obtain

$$\frac{dy}{dx}=\frac{\frac{d}{d\phi}(1-\mathrm{cos}\phi )}{\frac{d}{d\phi}(\phi -\mathrm{sin}\phi )}=\frac{\mathrm{sin}\phi}{1-\mathrm{cos}\phi}=\mathrm{cot}\frac{\phi}{2}.$$ |

Title | derivative for parametric form |
---|---|

Canonical name | DerivativeForParametricForm |

Date of creation | 2013-03-22 17:30:48 |

Last modified on | 2013-03-22 17:30:48 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Derivation^{} |

Classification | msc 26B05 |

Classification | msc 46G05 |

Classification | msc 26A24 |

Related topic | GoniometricFormulae |

Related topic | CurvatureOfNielsensSpiral |

Related topic | Parameter |