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

Compute the gradient of the function 𝑓(π‘₯, 𝑦) = π‘₯^(2)𝑒^(𝑦).

02:13

Video Transcript

Compute the gradient of the function 𝑓 of π‘₯ comma 𝑦 equals π‘₯ squared 𝑒 to the power of 𝑦.

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

So first of all, I’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, I’m gonna differentiate 𝑒 to the power of 𝑦 with respect to 𝑦. And, this is just gonna give us 𝑒 to the power of 𝑦 because this is one of our standard derivatives.

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

So, we’ve got π‘₯ squared multiplied by 𝑒 to the power of 𝑦 because 𝑒 to the power of 𝑦 was our derivative of 𝑒 to the power of 𝑦. So now, we’ve computed the gradient of the function 𝑓 of π‘₯ comma 𝑦 equals π‘₯ squared 𝑒 to the power of 𝑦. And, it is the gradient vector field two π‘₯𝑒 to the power of 𝑦 comma π‘₯ squared 𝑒 to the power of 𝑦.

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