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

Find the first partial derivative with respect to π‘₯ of the function 𝑓(π‘₯, 𝑦) = π‘₯Β² + 𝑦².

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Video Transcript

Find the first partial derivative with respect to π‘₯ of the function 𝑓 of π‘₯𝑦 equals π‘₯ squared plus 𝑦 squared.

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

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 π‘₯ 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 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 𝑦 squared. Well, we said we’re treating 𝑦 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 𝑓 of π‘₯𝑦 is π‘₯ squared plus 𝑦 squared. It’s two π‘₯.

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