### Video Transcript

Find the first partial derivative with respect to π₯ of the function π of π₯π¦ equals π₯ squared plus π¦ squared.

π of π₯π¦ is a multivariable function. Itβs a function in terms of both π₯ and π¦. The question is asking us to find the first partial derivative with respect to π₯ of this function. Thatβs often denoted as shown. This can also be pronounced as del π del π₯.

So, how do we find this first partial derivative? Well, what weβre really doing is looking at how the function changes as we just let the variable π₯ change and we hold the others constant. So, essentially, we differentiate term by term, but we treat π¦ itself as a constant. And so, we ask ourselves: whatβs the derivative of π₯ squared with respect to π₯?

Well, we know that to differentiate a power term, we multiply the entire term by the exponent and reduce that exponent by one. So, the derivative of π₯ squared is two π₯. But what about when we differentiate π¦ squared. Well, we said weβre treating π¦ in this case as a constant. And the derivative with respect to π₯ of a constant is zero. So, the first partial derivative with respect to π₯ of our function is two π₯ plus zero, or simply two π₯.

And we finish. We found the first partial derivative with respect to π₯ of the function π of π₯π¦ is π₯ squared plus π¦ squared. Itβs two π₯.