Let’s move on to our next type of equation. We are going to start looking at parametric equations.
For that, we're going to have to learn how to use parametric differentiation. Very simply put
parametric equations are just using a third variable. So, you will find these equations will have
either a t, or a p, or a z, or something that defines a third variable. So, you'll still have your x's and y's
and then you'll have the third variable that holds them together. That is known as a parameter.
So, that's where parametric equations come from. That's where parametric differentiation comes from.
It's very commonly seen in things like mechanics, so where things are changing over time, even in the field
of medicine. If you're measuring the rate of something over time, you will find that it is quite often
modeled as parametric equations. They would look in general like this. You have x as a function of t.
So, rather than x as a function of x’s or y’s, it will be a function of t. Then you have y as a function of t.
T would then be the parameter that holds them together, so it can bring them together.
When you see, we do some examples, you'll see how they work and how they come together
to form one function. It really is quite interesting. So, x and y are defined in terms of a third variable
sometimes, which is our parameter. Before we start our parametric differentiation calculations,
I feel like it’s a good point to recap over the fast method of differentiation. You've been doing this
all through. So, you’ve done this when we’re looking at chain rule, product rule, quotient rule.
We've just looked at implicit differentiation. We're now moving on to parametric.
But remember the basics are essential here, that if you have a function y equals to x to the power of n,
you bring the power down. You decrease the power by 1. And then, you will get the differential.
Now, this finds the gradient of a curve at a point. So, we have discussed previously that it would give you
a general gradient. Then you can find a particular gradient by just substituting your x or y value then.
We've seen us do this in the past few examples where we're differentiating implicitly
or we're finding gradients of tangents or gradients of normals.