Computing the partial derivative of a vector-valued function

Hello everyone. It's what I'd like to do here, and in the following few videos, is talk about how you take the partial derivative of vector-valued functions.

So the kind of thing I have in mind there will be a function with a multiple variable input. So this specific example has a two-variable input T and s. You could think of that as a two-dimensional space, as the input are just two separate numbers, and its output will be three-dimensional. The first component is T squared minus s squared. The Y component will be s times T, and that Z component will be T times s squared minus s times T squared minus s times T squared.

And the way that you compute a partial derivative of a guy like this is actually relatively straightforward. It's, if you were to just guess what it might mean, you’d probably guess right: it will look like the partial of V with respect to one of its input variables, and I'll choose T with respect to T. You just do it component-wise, which means you look at each component and you do the partial derivative to that because each component is just a normal scalar-valued function.

So you go up to the top one, and you say T squared looks like a variable as far as T is concerned, and its derivative is 2T. But s squared looks like a constant, so its derivative is zero. s times T, when s is s, looks like a constant, and when T looks like a variable, it has a derivative of s. Then T times s squared, when T is the variable and s is the constant, it just looks like that constant, which is s squared minus s times T squared.

So now, the derivative of T squared is 2T, and that constant s stays in. So that’s 2 times s times T, and that’s how you compute it probably relatively straightforward. The way you do it with respect to s is very similar. But where this gets fun and where this gets cool is how you interpret the partial derivative, right?

How you interpret this value that we just found, and what that means, depends a lot on how you actually visualize the function. So what I'll go ahead and do in the next video, and in the next few ones, is talk about visualizing this function. It'll be as a parametric surface in three-dimensional space; that's why I've got my graph or program out here. I think you'll find there's actually a very satisfying understanding of what this value means.