Second Derivatives of Parametric Equations

#9.2 Second Derivatives of Parametric Equations

Refresher time! Recall from 9.1 Defining and Differentiating Parametric Equations the following ideas:

1. Parametric functions are functions in which independent functions x and y are connected via t, a dummy variable representing time.
2. To calculate derivatives of parametric equations, $dy/dx$, we first find $dy/dt$ (from y(t)) and $dx/dt$ (from x(t)) and then divide the former by the latter.
3. Parametric equations are useful in determining the slope of a tangent line at a given point; more broadly speaking, they can inform us about rates of change of physical phenomena like motion.

As point (2) emphasized, we left 9.1 with the toolkit to calculate the first derivative of curves derived parametrically. This section, on the other hand, focuses on computing the second derivative of parametric equations we’ve been describing so far. 🧠

#🤔 Finding Second Derivatives of Parametric Equations

Put simply, we denote the second derivative of a parametric function as follows:

$$\frac{d^2y}{dx^2}$$

To find said derivative, we’ll use the chain rule: ⛓️

$$\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})=\frac{dy/dt}{dx/dt}=\frac{d}{dt}\frac{dt}{dx}(\frac{dy/dt}{dx/dt})=\frac{\frac{d}{dt}(\frac{dy/dt}{dx/dt})}{\frac{dx}{dt}}=\boxed{\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}}$$

#🔨 Breaking Down the Second Derivative

Here's a more in-depth description of the formula above:

Like in any other function, finding the second derivative of a parametric function involves taking the derivative of the first derivative of the function. In order to ...