How to solve partial derivative of f(x, y) = xy/(x^2 + y^2)?
Function of two variables, and we're going to find the partial derivatives with respect to X and with respect to Y. So, this is a hard problem. It looks like we're gonna have to use the Porsha rule. So, as a refresher, if you don't remember, the quotient rule says if you have F over G and you take the derivative, think of F as the top function and G as the bottom function. So, it's the derivative of the first (or the top piece) times the second (or the bottom piece) minus the first times the derivative of the bottom, second over the second squared. So, I think if this is first and this is second, since the derivative of the first times the second minus the first times the derivative of the second over the bottom of the second one squared. So, here when we take this derivative, we're gonna do the same thing.
Del F del X. This is our top piece and this is our bottom piece (or first or second). So, when we're taking the derivative with respect to X, we're fixing all of the Y's. So, here when we take this derivative, we're treating Y as constant. So, the derivative of X is one. So, that's gonna be what? That's the derivative of x? 1. It's the derivative of the first times the second. So far, so good. Minus the first. So, just XY times the derivative of the second. So again, here we're taking the derivative with respect to X. So, the derivative of Y squared is zero. This will just be 2x all over the second piece squared. Let's check the work on that just to make sure we didn't mess up.
The derivative of the first, when it's with respect to X, so the derivative of the first is one. Boom, and one cancels out, times a second minus the first times this derivative here, but this derivative is zero. So, you get 2x. Let's clean this up and see what happens. We get Y times x squared. So, we get Yx squared plus Y times y squared, which is y cubed, minus minus x squared minus x squared Y, all over this piece here. And then, what happens here? Oh, looks like these guys are like terms, right? Negative 2X squared Y plus Yx squared. There's a 1 here, so we're just going to get negative Yx squared, right? Because it's 1 minus 2 is negative 1. And then on the bottom, oh, plus y cubed. And then on the bottom, we just have this piece. So, it'd be x squared plus y squared squared. And that's the first partial derivative with respect to X. Hope that made sense. A harder example because you had to use the quotient rule.