Second-Order Derivatives
The second derivative of a function [latex]y = f(x)[/latex] is simply the derivative of the first derivative:
[latex]\frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{dy}{dx}\right][/latex]
For parametric equations, we already know that [latex]\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}[/latex]. To find the second derivative, we treat [latex]\frac{dy}{dx}[/latex] itself as a function that depends on [latex]t[/latex], then apply the chain rule:
[latex]\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}[/latex]
second derivative for parametric functions
If [latex]x = f(t)[/latex] and [latex]y = g(t)[/latex], then:
[latex]\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}[/latex]
Calculate the second derivative [latex]\frac{{d}^{2}y}{d{x}^{2}}[/latex] for the plane curve defined by the parametric equations [latex]x\left(t\right)={t}^{2}-3,y\left(t\right)=2t - 1,-3\le t\le 4[/latex].
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