Learn about Partial Derivatives 2 Video

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Learn about Partial Derivatives 2

Learn about Partial Derivatives 2 Before we move on to other functions let’s also take the partial derivative of our function here f(x, y) or the partial derivative of z with respect to y. So the partial derivative of z with respect to y, well now we’re saying how much the z change with respect to y if x is constant, so this x2 we treat it as constant now so the derivative of constant with respect to y is 0 so you ignore it. Now this x, y term the way we’re viewing it now is y is a variable x is a constant so what is the derivative of 5y with respect to y. Well it’s 5 so the derivative of xy with respect to y is just x and what is the derivative of y2 with respect to y, it’s just 2y. So you could see its quite symmetric. The partial of z with respect to x is 2x + y, the partial of z with respect to y is x + 2y and that’s because this equation is pretty symmetric the x’s and the y’s is kind of do the same thing. Now we pick to the point, x is =.2, y = .3 and actually let me erase this because I picked the different point where I graph it. I have to graph ahead of time just to see if I have to. So I don’t think I have to do include this anymore. So what I do is I picked the point x= .3, y= .3 and when x = .3, y = .3 what is z equal to. So .32 is .09, so z= .27. All right there is a substitute .3 and for x and y, z = .27. So what is the partial of z with respect to x at that point or we could write f so I have x at the point y. x = .3, y =.3. We figure that let’s say 2x .3= .6 + .3 = .9. So the slope in the x direction of that point is .9 and then if we take the partial with respect to y at that same point .3 + .6= .9. So let’s see if we can visualize this and bring in my graph, there we go. So this surface once again is the surface of z = x2 + xy +y2 and I just box this kind of the domain and the x and y dimensions that I defined kind of I bounded it because it starts to increase really fast and you wouldn’t be able to see all of these interesting stuff that happens closer in but what I did so this vertical line, I just wanted to show you that when x = .3, y = .3, z = .7 so that just kind of shows you that. That shows what point we’re working with and then these two lines, this is if you think about it, this is the line where y is constant, so this is the slope in the z or is the surface of changes with respect to x at this point. So this is the tangent line relative to x so we could kind of view it as. If we hold y constant here is the tangent line at that point and then if you hold x constant here is the tangent at that point and as I said in the last video you could actually have an infinite tangent lines. You have to pick the direction that you want to go in the x, y plane and then you could plot a tangent line. And so that’s why did partial derivatives to begin with and then actually this is pretty cool we can actually take and get my mouse, we can actually zoom in on this and zoom in a little bit more if you want to zoom in on the part that is interesting and let me translate this. So that’s the part that’s interesting and now what we rotate it so you can actually rotate so this is the tangent, this shows that the partial of the function with respect to y, the slope is .9 and this line shows that the partial of z or the partial of the function with respect to x is .9 at this point, at the point x is .3, y= .3 z .27 and we can rotate it just to get more intuition. The graphing thing looks a bit funny sometime but you see that both of those lines are tangent at that point and in fact two lines defined a plane and the plane that is defined by those lines or any of the two tangent lines to that point defines a tangent plane to the surface. So surface does have only one tangent plane but within a tangent plane there are an infinite number of tangent lines. Well anyway that’s the fun with graphing. Now let’s just chuck through a bunch of partial derivative problems just so that you get use to the mathematics on it. Let’s do some that might confuse you just so you see how you do them, let’s say that f(x, y) and I’m confining it to 3 dimensions although we can do it more actually and maybe I’ll do it more in more dimensions. Now we’re not going try to visualize it. Let’s say it’s x sin (x) cos (y) so let’s take the partial of f with respect to x. This is still going to be a function of x and y. So we treat y like a constant so if cosine of a constant this is just going to be a constant so we could almost ignore that we can put that out front. We could say that it’s going to be cosine of y times the derivative of this with respect to x, so you could say cosine of y. It’s just a number. This cosine of y could just be 5 or pi or whatever, cosine of y and then when you take the derivative of the constant just comes out of the derivative and then we take the derivative of the x’s so the derivative of the first term with respect to x well that’s just 1 times the 2nd expression so it’s the sin (x). I’m just doing the product rule here plus the derivative of the 2nd expression. That’s cos(x), cos(x) is the derivative of the 2nd expression times the 1st expression times x. So if we wanted to expand it all out the partial of f with respect to x is the function of x and y. It equals sin (x) cos (y) + x cos (x) cos(y).Not too difficult, you just have to realize it that anything with the y is a constant, so let’s reverse it. We’ll not reverse it let’s take a partial now and the y direction, how much is f change in the y direction, how much is f change in the y direction, if we hold x constant, so the partial of f with respect to y is still a function of x and y, the derivative in that direction is a function of x and y. So now x is a constant, so this actually becomes pretty shared for, this whole x sin (x) if x is some number 5 this is just the constant so we can just write that out front so that’s just x sin (x) and I know its hard for you to get used to saying that oh x sin of x just that a constant number because we’re still using the derivative with respect to x and that’s the hardest part about doing this partial derivatives but anyway this is just a constant term and I would just take the derivative of this with respect to y. The derivative of cos (y) with respect to y is –sin (y). There you have it. Let’s do another one and actually I’m going to add more variables just so you get used to the different notation x = a2 (b3) (c1/2). Now what is the partial derivative of x with respect to a. Well everything it also just a constant. What’s the partial of a2 with respect to a, what’s 2a so it will be just be 2a times the constant times b3, c1/2. I could actually get rid of parenthesis there. What’s the partial of x with respect to b? Well now a2 and c ½ are just constants, we could just write that a2 c1/2 and now we’ll just take the derivative of what is to be with 3b2 (3b2) and if I just want to rearrange it that’s 3a2 b2 c½. Not too difficult you just have to keep in mind what’s constant and what’s not and then finally the partial of x with respect to c, a2 b3. Those are both constant a2 b3 those are both constant a2 b3 times the derivative of this with respect to c½, c-1/2 or we could rewrite this as a2 b3/2 √c just a little bit of algebraic manipulation. Anyway, I will see you in the next video.

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Learn about Partial Derivatives 2

Before we move on to other functions let’s also take the partial derivative of our function here f(x, y) or the partial derivative of z with respect to y. So the partial derivative of z with respect to y, well now we’re saying how much the z change with respect to y if x is constant, so this x2 we treat it as constant now so the derivative of constant with respect to y is 0 so you ignore it. Now this x, y term the way we’re viewing it now is y is a variable x is a constant so what is the derivative of 5y with respect to y. Well it’s 5 so the derivative of xy with respect to y is just x and what is the derivative of y2 with respect to y, it’s just 2y.

So you could see its quite symmetric. The partial of z with respect to x is 2x + y, the partial of z with respect to y is x + 2y and that’s because this equation is pretty symmetric the x’s and the y’s is kind of do the same thing. Now we pick to the point, x is =.2, y = .3 and actually let me erase this because I picked the different point where I graph it. I have to graph ahead of time just to see if I have to. So I don’t think I have to do include this anymore. So what I do is I picked the point x= .3, y= .3 and when x = .3, y = .3 what is z equal to.

So .32 is .09, so z= .27. All right there is a substitute .3 and for x and y, z = .27. So what is the partial of z with respect to x at that point or we could write f so I have x at the point y. x = .3, y =.3. We figure that let’s say 2x .3= .6 + .3 = .9. So the slope in the x direction of that point is .9 and then if we take the partial with respect to y at that same point .3 + .6= .9. So let’s see if we can visualize this and bring in my graph, there we go.

So this surface once again is the surface of z = x2 + xy +y2 and I just box this kind of the domain and the x and y dimensions that I defined kind of I bounded it because it starts to increase really fast and you wouldn’t be able to see all of these interesting stuff that happens closer in but what I did so this vertical line, I just wanted to show you that when x = .3, y = .3, z = .7 so that just kind of shows you that. That shows what point we’re working with and then these two lines, this is if you think about it, this is the line where y is constant, so this is the slope in the z or is the surface of changes with respect to x at this point. So this is the tangent line relative to x so we could kind of view it as. If we hold y constant here is the tangent line at that point and then if you hold x constant here is the tangent at that point and as I said in the last video you could actually have an infinite tangent lines. You have to pick the direction that you want to go in the x, y plane and then you could plot a tangent line.

And so that’s why did partial derivatives to begin with and then actually this is pretty cool we can actually take and get my mouse, we can actually zoom in on this and zoom in a little bit more if you want to zoom in on the part that is interesting and let me translate this. So that’s the part that’s interesting and now what we rotate it so you can actually rotate so this is the tangent, this shows that the partial of the function with respect to y, the slope is .9 and this line shows that the partial of z or the partial of the function with respect to x is .9 at this point, at the point x is .3, y= .3 z .27 and we can rotate it just to get more intuition.

The graphing thing looks a bit funny sometime but you see that both of those lines are tangent at that point and in fact two lines defined a plane and the plane that is defined by those lines or any of the two tangent lines to that point defines a tangent plane to the surface.

So surface does have only one tangent plane but within a tangent plane there are an infinite number of tangent lines. Well anyway that’s the fun with graphing. Now let’s just chuck through a bunch of partial derivative problems just so that you get use to the mathematics on it. Let’s do some that might confuse you just so you see how you do them, let’s say that f(x, y) and I’m confining it to 3 dimensions although we can do it more actually and maybe I’ll do it more in more dimensions. Now we’re not going try to visualize it. Let’s say it’s x sin (x) cos (y) so let’s take the partial of f with respect to x. This is still going to be a function of x and y.

So we treat y like a constant so if cosine of a constant this is just going to be a constant so we could almost ignore that we can put that out front. We could say that it’s going to be cosine of y times the derivative of this with respect to x, so you could say cosine of y. It’s just a number. This cosine of y could just be 5 or pi or whatever, cosine of y and then when you take the derivative of the constant just comes out of the derivative and then we take the derivative of the x’s so the derivative of the first term with respect to x well that’s just 1 times the 2nd expression so it’s the sin (x). I’m just doing the product rule here plus the derivative of the 2nd expression. That’s cos(x), cos(x) is the derivative of the 2nd expression times the 1st expression times x.

So if we wanted to expand it all out the partial of f with respect to x is the function of x and y. It equals sin (x) cos (y) + x cos (x) cos(y).Not too difficult, you just have to realize it that anything with the y is a constant, so let’s reverse it. We’ll not reverse it let’s take a partial now and the y direction, how much is f change in the y direction, how much is f change in the y direction, if we hold x constant, so the partial of f with respect to y is still a function of x and y, the derivative in that direction is a function of x and y.

So now x is a constant, so this actually becomes pretty shared for, this whole x sin (x) if x is some number 5 this is just the constant so we can just write that out front so that’s just x sin (x) and I know its hard for you to get used to saying that oh x sin of x just that a constant number because we’re still using the derivative with respect to x and that’s the hardest part about doing this partial derivatives but anyway this is just a constant term and I would just take the derivative of this with respect to y.

The derivative of cos (y) with respect to y is –sin (y). There you have it. Let’s do another one and actually I’m going to add more variables just so you get used to the different notation x = a2 (b3) (c1/2). Now what is the partial derivative of x with respect to a. Well everything it also just a constant. What’s the partial of a2 with respect to a, what’s 2a so it will be just be 2a times the constant times b3, c1/2. I could actually get rid of parenthesis there.

What’s the partial of x with respect to b? Well now a2 and c ½ are just constants, we could just write that a2 c1/2 and now we’ll just take the derivative of what is to be with 3b2 (3b2) and if I just want to rearrange it that’s 3a2 b2 c½. Not too difficult you just have to keep in mind what’s constant and what’s not and then finally the partial of x with respect to c, a2 b3. Those are both constant a2 b3 those are both constant a2 b3 times the derivative of this with respect to c½, c-1/2 or we could rewrite this as a2 b3/2 √c just a little bit of algebraic manipulation. Anyway, I will see you in the next video.

Transcription by: Scribe4you Transcription Services