Video: Finding the First Partial Derivative of a Multivariable Function of Two Variables

Video Transcript

Find the first partial derivative with respect to 𝑥 of the function 𝑓 of 𝑥𝑦 equals 𝑥 squared plus 𝑦 squared.

𝑓 of 𝑥𝑦 is a multivariable function. It’s a function in terms of both 𝑥 and 𝑦. The question is asking us to find the first partial derivative with respect to 𝑥 of this function. That’s often denoted as shown. This can also be pronounced as del 𝑓 del 𝑥.

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

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 𝑥 squared is two 𝑥. But what about when we differentiate 𝑦 squared. Well, we said we’re treating 𝑦 in this case as a constant. And the derivative with respect to 𝑥 of a constant is zero. So, the first partial derivative with respect to 𝑥 of our function is two 𝑥 plus zero, or simply two 𝑥.

And we finish. We found the first partial derivative with respect to 𝑥 of the function 𝑓 of 𝑥𝑦 is 𝑥 squared plus 𝑦 squared. It’s two 𝑥.