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Show that the mixed second derivatives for the Cobb-Douglas Production function are equal.

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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01:51

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So we have the Cobb Douglas production function here, and we just need to find the mixed partial fractions of them are not partial fractions. Partial derivatives. Um, so let's go ahead and first find this with respect to, um X. So if we do that so remember this Why we assume is a constant to see is also a constant. So we can write this s o f sub X is equal to so see why race in the Beta del by Dell X of X to the Alfa and then taking the director of that We use power rule. So would be See why beta and then Alfa X to the Alfa minus one. Yeah, And now we can go ahead and do del by Dell. Why? To get our mixed partial. And now we're going to treat X is a constant. So we could go ahead and pull that out so it bc alfa X to the Alfa minus one and then del Beidle Why of why beta? And then we use power once again to move that out front. So this would be see Alfa Beta X Alfa minus one. Why? Beta minus? Well, so this is our first mixed partial and now we just need to take the derivatives in the opposite order. So coming up here, So instead of taking X first, we'll go ahead and take Why So let's get rid of all of that. So now we do, del by del y. So that would be f sub y. So we retreat C and X alfa as constant. So the c x, Alfa del Beidle Why of why To the beta, um, he is power real for that so moved the power out front. So just BC beta X alfa Why beta minus one? And then we can take the partial of this with respect to X and then we get our mixed partial. So the why we're going to treat is a constant. Also, we go in and pull that out and then who will have this here? And then again, we'll just go ahead and use power Will move. Alfa outfront Would BC Alfa Beta. Why? To the beta minus one x to the Alfa minus one and then I'll just actually go ahead and rearrange these attacks out front. And now, um, let me just pick this up. We can compare these two. Yeah, and we see that they are indeed the same. So those are equal. So we've shown that the Cobb Douglas production function does have where they're mixed second, partial derivatives.

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