### Video Transcript

Compute the gradient of the function π of π₯ comma π¦ equals π₯ squared π to the power of π¦.

So in this kinda question, what weβre looking for if weβre looking to compute the gradient is find the gradient vector field. And to do that, what weβre going to do is find the partial derivative of π₯ with π¦ as a constant, and then find the partial derivative of π¦ with π₯ as a constant. So, what weβre gonna do in real terms is differentiate π₯ squared with respect to π₯ and differentiate π to the power of π¦ with respect to π¦. And then as I said before, weβre also gonna include the constants, which the constants are gonna include the other terms.

So first of all, Iβm gonna differentiate π₯ squared with respect to π₯, which is equal to two π₯ because we multiply the exponent by a constant. So, two multiplied by one and then reduce the exponent by one. So, we get two π₯. And then, Iβm gonna differentiate π to the power of π¦ with respect to π¦. And, this is just gonna give us π to the power of π¦ because this is one of our standard derivatives.

So now, if weβre gonna find the gradient vector field of π of π₯ comma π¦, then what weβre gonna do for the first part, as I said, is the partial derivative of π₯ with π¦ as a constant. So, we have our two π₯ because that was our partial derivative of the π₯ term. And then, our term, π to the power of π¦, is just our constant. So, we get two π₯π to the power of π¦. And then, we have our partial derivative of π¦ multiplied by our constant containing π₯.

So, weβve got π₯ squared multiplied by π to the power of π¦ because π to the power of π¦ was our derivative of π to the power of π¦. So now, weβve computed the gradient of the function π of π₯ comma π¦ equals π₯ squared π to the power of π¦. And, it is the gradient vector field two π₯π to the power of π¦ comma π₯ squared π to the power of π¦.