For f(x, y, z)=4 x^3 y^2 z^2+4 x^2+2 y^3+5 z^5+3 x-2 y+11 z+12 find fx z y(x, y, z) and fx y z(x

For f(x, y, z)=4 x^3 y^2 z^2+4 x^2+2 y^3+5 z^5+3 x-2 y+11...

Key Concepts

Partial Differentiation

Partial differentiation involves computing the derivative of a multivariable function with respect to one variable while treating all other variables as constants. This is a fundamental concept in multivariable calculus used to analyze how functions change along different axes.

Mixed Partial Derivatives

Mixed partial derivatives are the result of taking partial derivatives of a function with respect to different variables in succession. They measure how the derivative with respect to one variable changes as another variable is varied, providing insight into the interplay between variables in a multivariable function.

Clairaut's Theorem

Clairaut's theorem states that if the mixed partial derivatives of a function are continuous around a point, then the order in which the derivatives are taken does not matter, meaning that the mixed derivatives are equal. This theorem justifies the equality of different mixed derivative orders for well-behaved functions.

Transcript

00:01 So they want us to find the partials of f of x, y, z, and then of f x zy, y, but we actually have a theorem in this chapter that says, it doesn't really matter what order we do these in, especially if these are all going to be kind of continuous in some sense, or smooth functions.

00:25 And so this is essentially kind of like a three -variable polynomial in some sense.

00:31 Sense, this will always be smooth.

00:34 So these two will be equal to each other.

00:38 So if we find one of them, then we've actually found the other.

00:43 So i'm just going to do it in this first case here.

00:48 So let's just go ahead and do that.

00:50 So it will be del by del x to start.

00:56 And so remember, we're going to assume all of the variables other than x are a constant.

01:04 So when we're looking through this, this y cube times 2, well, the derivative of that would be 0, because we're assuming it's a constant.

01:12 Same thing with z to the 5th times 5, 2 times y, 11 times z, and then 12 is also just going to be 0.

01:23 And then we can just factor out this y squared z squared, just like we put the 4 when we take the partial of that, and then just take the derivatives of everything else.

01:32 So let's just go ahead and write that out.

01:33 So f sub x is going to be, so i'll actually just write this out first.

01:39 So 4, y squared, c squared, and then del by del x of x cubed.

01:45 And then the derivative of 4x squared, we would use power rule.

01:49 So 8x, derivative of 3x would just be 3.

01:54 And then taking the derivative, that would be 3x squared.

02:01 So we can go ahead and multiply all of that together now...

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