{"id":994,"date":"2025-06-20T17:27:15","date_gmt":"2025-06-20T17:27:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=994"},"modified":"2025-08-20T16:38:27","modified_gmt":"2025-08-20T16:38:27","slug":"calculus-with-parametric-curves-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/calculus-with-parametric-curves-learn-it-2\/","title":{"raw":"Calculus with Parametric Curves: Learn It 2","rendered":"Calculus with Parametric Curves: Learn It 2"},"content":{"raw":"

Second-Order Derivatives<\/h2>\r\n

The second derivative of a function [latex]y = f(x)[\/latex] is simply the derivative of the first derivative:<\/p>\r\n

[latex]\\frac{d^2y}{dx^2} = \\frac{d}{dx}\\left[\\frac{dy}{dx}\\right][\/latex]<\/p>\r\n

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:<\/p>\r\n

[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]<\/p>\r\n\r\n

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second derivative for parametric functions<\/h3>\r\n

If [latex]x = f(t)[\/latex] and [latex]y = g(t)[\/latex], then:<\/p>\r\n

[latex]\\frac{d^2y}{dx^2} = \\frac{\\frac{d}{dt}\\left(\\frac{dy}{dx}\\right)}{\\frac{dx}{dt}}[\/latex]<\/p>\r\n\r\n<\/section>

When you know [latex]\\frac{dy}{dx}[\/latex] as a function of [latex]t[\/latex], this formula becomes straightforward to apply\u2014just differentiate with respect to [latex]t[\/latex] and divide by [latex]\\frac{dx}{dt}[\/latex].\r\n\r\n<\/section>
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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].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558893\"]\r\n

\r\n\r\nFrom the example: Finding the Derivative of a Parametric Curve we know that [latex]\\frac{dy}{dx}=\\frac{2}{2t}=\\frac{1}{t}[\/latex]. Using our above equation, we obtain\r\n
[latex]\\frac{{d}^{2}y}{d{x}^{2}}=\\frac{\\left(\\frac{d}{dt}\\right)\\left(\\frac{dy}{dx}\\right)}{\\frac{dx}{dt}}=\\frac{\\left(\\frac{d}{dt}\\right)\\left(\\frac{1}{t}\\right)}{2t}=\\frac{-{t}^{-2}}{2t}=-\\frac{1}{2{t}^{3}}[\/latex].<\/div>\r\n \r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>
Watch the following video to see the worked solution to the example above.