# Video: Finding the Gradient of a Multivariable Function of Two Variables

Compute the gradient of the function π(π₯, π¦) = π₯^(2)π^(π¦).

02:13

### 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 weβre looking for if weβre looking to compute the gradient is find the gradient vector field. And to do that, what weβre 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 weβre 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, weβre also gonna include the constants, which the constants are gonna include the other terms.

So first of all, Iβm 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, Iβm 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 weβre gonna find the gradient vector field of π of π₯ comma π¦, then what weβre 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, weβve 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, weβve 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¦.

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