2/3/2024 0 Comments Define gradientJust be two of them, and this is kind of a weird thing because it's like what, this is a vector, it's So the nabla symbol is this vector full of different partial derivative operators. So you give this guy the function f and it gives you this expression, this multi-variable function as a result. And by operator, I just mean like partial with respect to x, something where youĬould give it a function, and it gives you another function. Think about this triangle, this nabla symbol as being a vector full of partial derivative operators. So a very helpful mnemonic device with the gradient is to Gradient as the full derivative cuz it kind of capturesĪll of the information that you need. Partial of f with respect to x, and partial of f with respect to y. Write this more generally is we could go down hereĪnd say the gradient of any function is equal to a vector with Partial derivatives, and a three-dimensional output. So you could also imagine doing this with three different variables. This is a function that takes in a point in two-dimensional space and outputs a two-dimensional vector. Little bit more room here and emphasize that it's got an x and a y. And notice, maybe I should emphasize, this is actually a vector-valued function. And the bottom one, partialĭerivative with respect to y X-squared cosine of y. So the first one is the partial derivative with respect to x, to x times sine of y. And what this equals is a vector that has those two The name of that symbol is nabla, but you often just pronounce it del, you'd say del f or gradient of f. I'll change colors here, you denote it with a little So now what the gradient does is it just puts both of these together in a vector. Now we look up here and we say x is considered a constant so x-squared is also considered a constant so this is just aĬonstant times sine of y, so that's gonna equal that same constant times the cosine of y, which is the derivative of sine. Whereas the partial derivative with respect to y. As far as x is concerned, the derivative of x is 2x so we see that this will be 2x times that constant sine of y, sine of y. So partial of f with respect to x is equal to, so we look at this and weĬonsider x the variable and y the constant. The gradient is a way of packing together all the partial derivative And let's say it's f of x, y,Įquals x-squared sine of y. And I'm just gonna make itĪ two-variable function. So on the computation side of things, let's say you have some sort of function. But to do that, we need to know what both of them actually are. We'll connect them in the next few videos. Unrelated to the intuition and you'll see that. Of those weird things where the way that you compute it actually seems kind of Go the other way around, but the gradient is one And I hate doing this, I hate showing the computation before the geometric intuition since usually it should And in this video, I'm only gonna describe how you compute the gradient, and in the next couple ones I'm gonna give the
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