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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Parametric derivative ： ウィキペディア英語版 Parametric derivative Parametric derivative is a derivative in calculus that is taken when both the ''x'' and ''y'' variables (traditionally independent and dependent, respectively) depend on an independent third variable ''t'', usually thought of as "time". ==Example== For example, consider the set of functions where: :$x(t)\; =\; 4t^2\; \backslash ,$ and :$y(t)\; =\; 3t.\; \backslash ,$ The first derivative of the parametric equations above is given by: :$\backslash frac\}\}\; =\; \backslash frac,$ where the notation $\backslash dot(t)$ denotes the derivative of ''x'' with respect to ''t'', for example. To understand why the derivative appears in this way, recall the chain rule for derivatives: :$\backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac,$ or in other words :$\backslash frac\; =\; \backslash frac\}\}.$ More formally, by the chain rule: $\backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac$ and dividing both sides by $\backslash frac$ gets the equation above. Differentiating both functions with respect to ''t'' leads to :$\backslash frac\; =\; 8t$ and :$\backslash frac\; =\; 3,$ respectively. Substituting these into the formula for the parametric derivative, we obtain :$\backslash frac\; =\; \backslash frac\}\; =\; \backslash frac,$ where $\backslash dot$ and $\backslash dot$ are understood to be functions of ''t''. The second derivative of a parametric equation is given by : |$=\; \backslash frac\backslash left(\backslash frac\backslash right)$ |- | |$=\backslash frac\backslash left(\backslash frac\backslash right)\backslash cdot\backslash frac$ |- | |$=\; \backslash frac\backslash left(\backslash frac\}\backslash right)\backslash frac\backslash ddot\; -\; \backslash dot\backslash ddot\}$ |} by making use of the quotient rule for derivatives. The latter result is useful in the computation of curvature. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Parametric derivative」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.