Instantly calculate the first derivative ($\frac{dy}{dx}$) and the second derivative ($\frac{d^2y}{dx^2}$) for equations defined parametrically by $x(t)$ and $y(t)$.
Calculating $\frac{dx}{dt}$, $\frac{dy}{dt}$, and applying the quotient rule for the second derivative...
Parametric equations define coordinates $(x, y)$ in terms of a third variable, usually $t$ (often representing time). Finding the derivative $\frac{dy}{dx}$ and the second derivative $\frac{d^2y}{dx^2}$ requires special formulas that rely on the Chain Rule.
The slope of a parametric curve, $\frac{dy}{dx}$, is found by taking the derivative of $y$ with respect to $t$ and dividing it by the derivative of $x$ with respect to $t$. This is derived directly from the Chain Rule.
The second derivative, which helps determine concavity, is calculated by differentiating the first derivative ($\frac{dy}{dx}$) with respect to $t$, and then dividing that result by $\frac{dx}{dt}$. It's a common mistake to forget the final division by $\frac{dx}{dt}$!
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