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Computing the partial derivative of a vector-valued function

When a function has a multidimensional input, and a multidimensional output, you can take its partial derivative by computing the partial derivative of each component in the output. Created by Grant Sanderson.

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Video transcript

- [Voiceover] Hello, everyone. So 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 will be a function with a multi-variable input, so this specific example have a two variable input, p and s. You could think of that as a two-dimensional space as the input or just two separate numbers. And its output will be three-dimensional. The first component, p 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 straight-forward. If you're to just guess what it might mean, you'll probably guess right. It will look like partial of v with respect to one of its input variables, and I'll choose t with respect to t. And you just do it component-wise, which means you look at each component and you with a partial derivative to that 'cause each component is just a normal scaler valued function. So you go up to the top one and you say t-squared looks like a variable, as far t is concerned, and this derivative is 2t. But s-squared looks like a constant, so its derivative is zero. S times t, when s looks like a constant and when t looks like a variable, has a derivative of s. Then t times s-squared, when t's the variable and s is the constant, just looks like that constant, which is s-squared minus s times t-squared. So now a derivative of t-squared is 2t and that constant s stays in. So that two 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 and three-dimensional space. That's why I got my grapher program out here and I think you'll find there's actually a very satisfying understanding of what this value means.