http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx(adsbygoogle = window.adsbygoogle || []).push({});

On this page the author makes it very clear that:

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

provided ##\frac{dx}{dt} \neq 0##.

In example 4, ##\frac{dx}{dt} = -2t##, which is zero when ##t## is zero. In simplifying ##\frac{dy}{dx}## the author even divides the numerator and denominator by ##t## which is only possible if ##t \neq 0##.

This is all consistent with the requirement ##\frac{dx}{dt} \neq 0##.

The author then obtains an expression for the second derivative in terms of ##t##, plugs in zero, and finds out that the second derivative is zero at ##t = 0##.

How is this consistent with the assumption that ##\frac{dx}{dt} \neq 0##? What's going on here?

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# Second derivative with parametric equations

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