Parametric Derivatives - Calculus 2

When working with plane/ parametric curves, it may sometimes be of interest to know the slope at a particular point along that curve. We learn in Calc 1 that the slope at a point along a curve/function is represented by the derivative of that function, and we have a whole arsenal of different differentiation rules that we can use to find a derivative for any particular function. So, in order to know the slope at a point along a parametric curve, we first need to know how to find the derivative for that curve. However, we will not be able to find this derivative, called the parametric derivative, the same way we have found derivatives for any other functions in the past.

Since plane/ parametric curves are defined with a set of parametric equations rather than one single equation, the process of finding the derivative will be different. As it turns out, the parametric derivative can be found by first differentiating each parametric equation with respect to t, and then dividing the derivative of the equation for y by the derivative of the equation for x (more details on why this is the case in the lesson video). So, finding parametric derivatives isn't all that difficult of a new concept if you are already familiar with the various differentiation rules and how to simplify rational expressions. And just like with regular derivatives for functions in terms of x and y, we can also find higher-order derivatives for sets of parametric equations. Each higher-order derivative involves differentiating the previous derivative with respect to t and then dividing it by the derivative of the original parametric equation for x.

We take a look at where these formulas come from and several examples of using them in this new lesson for Calculus 2, so let's get right into it!

In this lesson, you will learn:

• How to find the parametric derivative for a set of parametric equations
• How to find the second parametric derivative for a set of parametric equations
• How to find the third and other higher-order derivatives for a set of parametric equations

I hope you find this video to be helpful!

-Josh