Partial derivatives
Partial Derivatives 2 More on partial derivatives
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- Before we move on to other functions, let's also take the
- partial derivative of our function here, f of xy, or the
- partial derivative of z with respect to y.
- So let's do it in magenta.
- So the partial derivative of z with respect to y.
- Well, now we're saying how much does z change with respect
- y if x is constant.
- So this x squared we treat as a constant now.
- So the derivative of a constant with respect to
- y is 0, so ignore it.
- Now this xy term.
- The way we're doing it now is, y is a variable,
- x is a constant.
- So what is the derivative of, I don't know, 5y
- with respect to y?
- Well it's 5.
- So the derivative of xy with respect to y is just x.
- And what's the derivative of y squared with respect to y?
- Well it's just 2y.
- So you can see it's quite symmetric.
- The partial of z with respect to x is 2x plus y, the
- partial of z with respect to y is x plus 2y.
- That's because this equation is pretty symmetric.
- The x's and the y's kind of do the same thing.
- Now, we picked the point x is equal to 0.2 y is equal to 0.3.
- Actually let me erase this, because I picked a different
- point where I graph, and I graphed it ahead of time
- just to save time.
- So I don't think I have to include this anymore.
- So what I did, I picked the point x is equal to
- 0.3, y is equal to 0.3.
- And when x is equal to 0.3, y is equal to 0.3,
- what is z equal to?
- Let's see, 0.3 squared is 0.09, 0.3 times 0.3 0.09.
- So it's z is equal to 0.27, right?
- Just substitute 0.3 in for x and y.
- z is equal to 0.27.
- So what is the partial of z with respect
- to x at that point?
- Or we could write, f sub x at the point y.
- x is equal to 0.3, y is equal to 0.3.
- It equals, we figured that, let's see 2 times 0.3 is 0.6,
- plus 0.3, that's equal to 0.9.
- So the slope in the x direction at that point is 0.9.
- And if we take the partial with respect to y at that same
- point, 0.3 plus 0.6, that's also equal to 0.9.
- Let's see if we can visualize this.
- Let me bring in my graph.
- There we go.
- So this is, this surface once again is the surface of z is
- equal to x squared plus xy plus y squared.
- And this box is kind of the domain, and the x and y
- dimensions that I define.
- Kind of, I bounded it because it starts to increase really
- fast, and you wouldn't be able to see all this interesting
- stuff that happens closer in.
- But this, what I did, so this vertical line, I just wanted
- to show you that when x is equal to 0.3, y is equal
- to 0.3, z is equal to 0.7.
- So that just kind of helps show you that, ok, 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, right?
- So this is the slope in the z, or as the surface changes
- with respect to x at this point, right?
- This is the tangent line relative to x.
- So you could kind of view it as, if you hold y
- constant, here's a tangent line at that point.
- And if you hold x constant, here's a tangent
- line at that point.
- And like I said in the last video, you can actually have
- infinite tangent lines.
- You have to pick the direction that you want to go in the xy
- plane, and then you could plot a tangent line.
- And so that's why we did partial derivatives
- to begin with.
- Actually, this is pretty cool.
- We can actually take, we can actually zoom in on this.
- Zoom it a little bit more.
- I want to zoom in on the part that is interesting,
- let me translate this.
- So that's the part that's interesting.
- And now let me 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 0.9.
- And this line shows that the partial of the slope, or the
- partial the function, sorry the partial of z, or the partial of
- the function with respect to x is 0.9, at this point.
- At the point x is 0.3, y is 0.3, right? x is 0.3, y is
- 0.3, z is equal to 0.27.
- And we can rotate it just to get more intuition.
- I think it's, the graphing thing looks a little
- bit funny sometimes.
- But you see that both of those lines are
- tangent at that point.
- And in fact, two lines define a plane, and the plane that's
- defined by those two lines, or any of the two tangent lines to
- that point, defines a tangent plane to the surface.
- So a 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 chug through a bunch of partial derivative
- problems just so that you get used to the mathematics of it.
- Delete that.
- Let's do some that might confuse you, so you
- see how to do them.
- Let's say that f of xy, and I'm confining it
- to three dimensions.
- Although we can do it more.
- Actually, maybe I'll do it more dimensions now that we're not
- going to try to visualize it.
- Let's say it's x sine of x, cosine of 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 cosine of a constant, this
- is just going to be a constant.
- So we can almost ignore that.
- We could 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, I don't know,
- 5 or pi, or whatever.
- Cosine of y.
- And then, because when you take the derivative the constant
- just comes out of the derivative.
- And then we would 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 second expression.
- So sine of x, I'm just doing the product rule here.
- Plus the derivative of the second expression,
- that's cosine of x.
- Cosine of x is the derivative of the second expression, times
- the first expression, times x.
- So If we wanted to expand it all out, the partial of f
- with respect to x, this is the function of x and y.
- It equals sine of x, cosine of y, plus, let's put this x out
- front, just so we change the order. x, cosine of
- x, cosine of y.
- Not too difficult.
- You just have to realize that anything with
- the y is a constant.
- So let's reverse it.
- Well, not reverse it.
- Let's take the partial now in the y direction.
- How much does f change in the y direction if
- we hold x constant.
- So the partial of f with respect to y, 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 straightforward.
- This whole x, sine of x, if x is some number, 5,
- this is just a constant.
- So we can just write that out front.
- So that's just x, sine of x.
- I know it's hard for you to get used to saying that, oh, x sine
- of x, that's just a constant number.
- Because you're so used to taking the derivative
- with respect to x.
- And that's the hardest part about doing these
- partial derivatives.
- But anyway, this is just a constant term.
- And now we just take the derivative of this
- with respect to y.
- The derivative of cosine of y with respect to
- y is minus sine of y.
- I'll do that in yellow.
- Minus sine of y.
- I just want to put the minus out front.
- Minus sine of y.
- There you have it.
- Let's do another one.
- And actually I'm going to add more variables.
- Let's say that, I don't know, just so you get used to
- the different notation.
- x is equal to a squared times, I don't know, a squared times b
- to the third, times c to the 1/2 power.
- Now what is the partial derivative of x
- with respect to a?
- Well everything else is just a constant.
- What's the partial of a squared with respect to a?
- Well it's 2a.
- So it'll just be 2a times the constants.
- Times b to the third, c to the 1/2.
- I can actually get rid of my parentheses there.
- What's the partial derivative with respect to b?
- Ah, sorry, what's the partial of x with respect to b?
- Well now, a squared and c to the 1/2 are just constants.
- We can just write that. a squared, c to the 1/2.
- And now we just take the derivative with respect to b.
- Well that's 3b squared.
- Times 3b squared.
- If I just want to rearrange it, that's 3a squared, b
- squared, c to the 1/2 power.
- 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.
- a squared, b to the third, those are both constants.
- a squared, b to the third times derivative of
- this with respect to c.
- 1/2 c to the minus 1/2.
- Or we could rewrite this as a squared, b to the third,
- over 2 square root of c.
- Just a little bit of algebraic manipulation.
- Anyway, I will see you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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