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Second derivative formula for curve defined by parametric equations

February 19, 2008

Suppose $x$ and $y$ are defined parametrically by

$\displaystyle x = \varphi (t) \text{ and } y = \psi (t).$

Show that

$\displaystyle \frac{d^2 y}{dx ^2} = \frac{\varphi '(t) \psi ''(t) - \varphi''(t) \psi '(t)}{\varphi ' (t)^3}.$

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1 Comment | Calculus, Differential equations | Tagged: Calculus, Differential equations, higher order derivative, parametric equation, second derivative | Permalink
Posted by Ernie

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