Derivative of a parametric function

So what we have here is X being defined in terms of T, and Y being defined in terms of T. Then, if you were to plot over all of the T values, you get a pretty cool plot just like this. So, you know, you try T equals z, figure out what X and Y are; T is equal to one, figure out what X and Y are; and all the other T's, and then you get this pretty cool looking graph.

But our goal in this video isn't just to appreciate the coolness of graphs or curves defined by parametric equations. We actually want to do some calculus, and in particular, we want to find the derivative. We want to find the derivative of y with respect to X, the derivative of y with respect to X when T is equal to 13.

If you are inspired, I encourage you to pause and try to solve this, and I am about to do it with you in case you already did or you just want me to. Alright, so the key is, well, how do you find the derivative with respect to X?

The derivative of y with respect to X, when they're both defined in terms of T, and the key realization is the derivative of y with respect to X is going to be equal to the derivative of y with respect to T over the derivative of x with respect to T. If you were to view these differentials as numbers, well, this would actually work out mathematically. Now, it gets a little bit non-rigorous when you start to do that, but if you thought of it that way, it’s an easy way of thinking about why this actually might make sense.

The derivative of something versus something else is equal to the derivative of y with respect to T over x with respect to T. Alright, so how does that help us? Well, we can figure out the derivative of x with respect to T and the derivative of y with respect to T.

The derivative of x with respect to T is just going to be equal to, let's see, the derivative of the outside with respect to the inside. It's going to be 2s—whoops, derivative of s is cosine 2 cosine of 1 + 3T times the derivative of the inside with respect to T. So that's going to be the derivative of one, which is just zero, and the derivative of 3T with respect to T is three, so times 3, that's the derivative of x with respect to T.

I just used the chain rule here, derivative of the outside to s of something with respect to the inside. So, derivative of this outside two s of something with respect to 1 + 3T is that right over there, and the derivative of the inside with respect to T is just R3.

Now, derivative of y with respect to T is a little bit more straightforward. The derivative of y with respect to T we just apply the power rule here: 3 * 2 is 6t to the 3 - 1 power, 6t^2. So this is going to be equal to 6t^2 over, well, we have the 2 * 3, so we have 6 * sine of 1 + 3T.

Then our sixes cancel out, and we are left with t^2 over sine of 1 + 3T. If we care when T is equal to 13, when T is equal to -1/3, this is going to be equal to, well, this is going to be equal to 1/3 times 2ar - 1/3 squared over the cosine of 1 + 3 * 1/3.

Is 1, so it's 1 + 1, so it's the cosine of 0, and the cosine of 0 is just going to be one. So this is going to be equal to positive, positive 1.

Now, let’s see if we can visualize what's going on here. So let me draw a little table here. I’m going to plot, I’m going to think about T, X, and Y. So T, X, and Y.

When T is equal to -13, well, our X is going to be, this is going to be sine of zero, so our X is going to be zero, and our Y is going to be, what, -2 over 27? So we're talking about the point (0, -2 over 27).

So that is that point right over there, so that's the point where we're trying to find the slope of the tangent line, and it's telling us that that slope is 1. So, if we move—I guess one way to think about it is, if we move 4, 1, 2, 3, 4 and half, we're going to move up half.

So if I wanted to draw the tangent line right there, it would look something like that—something, something, something like that. Let’s see—if we go one, two, three, four, and a half, so yeah, just like that. It’s pretty close.

So that's what we just figured out. We figured out that the slope of the tangent line right at that point is 1. So it's not only neat to look at, but I guess somewhat useful.