WEBVTT



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The directions are to differentiate.

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What does that mean in calculus, differentiate?

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All that means is to find the derivative.

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Find the derivative.

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And the answer is...

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8 u to the 7th.

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And what do I put on the left side?

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Y prime.

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However, I want to use Leibniz notation.

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That would be dy over du.

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Why is it dy over du?

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as the representation with respect to u.

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But what I didn't tell you is u is some function of x.

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And I really want dy dx, not dy du.

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Okay, so dy dx, I don't see any x's here.

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Okay, so we assume this variable is some function of x.

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Right now y is a function of u.

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So if I want dy dx, I go 8u to the 7th, and then by the chain rule, we multiply by

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the

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derivative of u with respect to x.

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This is 8u to the seventh times du over dx.

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The derivative of u with respect to x is cosine x.

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In terms of x, the result is 8 sine to the seventh x times cosine x.

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cosine of x.

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What if the directions asked us to find dy over dt?

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Okay. This implies, right? We only see y's and u's here. But if we're trying to find

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this, this

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implies that every variable is some function of t, but you just don't even know what

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it is. And

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we don't even care. So all we do here is

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the derivative of this function is dy with respect to t. We differentiate this 8u to

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the seventh and then by the chain rule we multiply by the derivative of this

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function.

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du over dt.

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Okay. You're done.

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Because we don't even know what U is in terms of T.

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We don't...

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Who cares?

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So we can take the derivative with respect to any variable.

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Even if we don't see the variable.

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you take the derivative of whatever this function is and then by the chain rule

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we take the derivative of this function u, since it's just u it's times derivative

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of u with

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This process is called implicit differentiation.

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What's this in that equation?

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A circle of radius?

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3.

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How do I know it's 3?

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It's r squared.

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squared. Okay. And where's the center of the circle?

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Oh, it's, um, zero, zero.

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Zero, zero. Okay. Do you remember this from a couple years ago?

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No. No? Okay.

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The equation of a circle is (x-h) squared plus (y-k) squared equals r squared.

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squared plus y minus k

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squared equals r squared. Is the equation

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have a circle of radius r centered at the e, okay. Okay, anyways, this is a circle

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centered at the origin of radius n. Is this a function? No. Why is this not a

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function? It

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fails the vertical line test. Okay, but still we call it in calculus called a

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curve. Okay, we don't care if it's not a function. Calculus doesn't really care

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about functions, whether it's a function or not. Okay, this is some curve. It has to

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be a

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We could solve for y: y squared is 9 minus x squared.

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Then y is plus or minus the square root of 9 minus x squared.

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which is two separate functions now this is a function this is a function okay

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The top semicircle is y equals the square root of 9 minus x squared.

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The bottom semicircle is y equals negative the square root of 9 minus x squared.

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Okay.

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Okay, is y a function of x here?

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It's not. For it to be a function of x, you have to have y equals something. This

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here, y is a function of x.

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Okay, Y is explicit. Y is explicit.

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A function of X.

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Here,

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We can imply it's a function of x, but we don't know.

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Okay, but here it's explicitly it's written y is explicitly some function of x here

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it might be might not we don't really care actually but.

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But we can apply it is even though it's not written it want as long.

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So anytime we have y equals some function, this is like explicit, and then here we

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can

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imply it is or it's not, so on. So we're gonna differentiate this even though y is

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not explicitly

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a function. Okay, and here's how we do it. The x squared plus y squared equals 9.

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We're

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to differentiate both sides of the equation not just the right side although we have

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been doing

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that. We just haven't really been talking about that. Okay. What's it like here?

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We just said this equals this before we went to you. I told you that. We actually

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are

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The derivative of y with respect to x is dy over dx.

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The derivative of 8u to the eighth is 8u to the seventh times du over dx.

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this kind of makes sense so we're gonna do the same thing except now we have

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variables on both sides so we just use all the same rules on the left hand side

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we have a sum. So how do I take the derivative of a sum? It's just the sum of

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each derivative. Okay so we start with this one. Derivative of x squared is 2x,

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except x is, could be a function of any variable we want. So we do need, I should

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We are differentiating with respect to x.

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with respect to x so it's 2x times the derivative of x with respect to x dx dx

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The derivative of y squared is 2y times dy over dx.

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dy dx

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we're done with the left hand side

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and the derivative of 9 is

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0 we don't need any chain rule because there's no variables

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okay and what is dx dx

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1 the rate of change of x

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with respect to the change of x is one. The rate of change is the same. So that's

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one.

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You can take it as a fraction as well. We get this.

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Okay, and if I want to find dy dx, we now just solve for this.

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Two y times dy over dx equals negative 2x, so divide by 2y.

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So what does this tell us?

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Okay.

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That's not a very good circle, but if we want to find the slope of the tangent line,

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which

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is our dy dx, it is our x and y.

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Okay.

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Okay.

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At x equals 2.

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So find the slope.

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So I need to plug in x equals 2 into here.

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At x equals 2, the derivative is negative 2 over y; we still need the y-coordinate.

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What's that?

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So this is a little this is a little weird because we normally don't do this very

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often. We need a y coordinate just like we plugged in an x coordinate. So how do I

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find

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this y coordinate right here? The original function right? That's this. Okay, so x

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squared

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At x equals 2, y squared equals 5.

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Thus y equals plus or minus the square root of 5.

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Yeah, when x is 2, we got one value here and we got one value here. So this will

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The two points are (2, square root of 5) and (2, negative square root of 5).

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The two slopes are negative 2 over square root of 5 and positive 2 over square root

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of 5.

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is going to be dy dx of 2, square root of 5 is negative 2 over square root of 5 and

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the other one

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So the slope of this line is exactly negative 2 over square root of 5 and the slope

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of this line is

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positive 2 over square root of 5.

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The harder method is to solve for y first and then differentiate.

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So dy dx would be what?

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That's 9 minus X squared.

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The 1 hat.

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And then 1 half times the derivative of the inside, which is negative 2x.

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And then if we wanted at x equals 2, we get 1 half 9 minus 2 squared 1 half times

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negative

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two times two

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And if you do a bunch of work comes out to be

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negative two over

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Five or something like that

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And we would only gotten one of the tangent lines slope to the tangent lines instead

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of two of them

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Then we have to do the same thing with the other one

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Okay, a lot of functions. There's no way to

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Change it to make it explicitly

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Okay

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Now take the same curve and find dy over dz.

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Now let's find d y d z I

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Don't even see these

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And we don't care. This implies that every variable is a function and it's a

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function of z.

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Okay. So we just start on the left hand side, differentiate the whole left hand

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side, which again is a sum.

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So we take the derivative of each piece separately.

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We get 2x times dx over dz.

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chain rule we multiply by the derivative of that function with respect to whatever

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variable.

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Plus 2y times dy over dz.

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to the variable z.

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And the derivative of constant is 0.

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And then we just solve for

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dy dz, which is this one right here.

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So we have two while.

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And then what those simplify and that's it.

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You will still have a, you have this rate of change in our answer and that's okay.

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We can differentiate with respect to any variable we want.

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If we are asked to find the slope of a tangent line, that's always dy ds.

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But if they just say find dy dz, that's all we do.

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Yeah, these are actually.

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Okay, this is definitely not a function.

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Who cares? Some curve.

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Let's see what it looks like.

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In fact, just for fun.

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The left side is sine x times e to the y.

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What did I say? Y cubed? That's kind of cool. Definitely not a function, but who

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cares?

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It's occurred. Okay, if we wanted to find the two slopes of the tangent lines at x

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equals 2,

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we're gonna have to differentiate. We'd have to plug it, figure out those y-values.

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It's not the point.

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Okay, let's find

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dy dx.

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Okay, we just differentiate the left hand side separately from the right hand side.

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So on the left hand side, what do we have here?

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A product.

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A product.

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So what rule do we got each?

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A product rule.

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Okay.

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Okay.

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Use the product rule: the derivative of sine x times e to the y, plus sine x times

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the derivative of e to the y.

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t to the y,

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sine t to the y.

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The derivative of sine x is cosine x, and the derivative of e to the y is e to the y

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times dy over dx.

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derivative of x with respect x times e to the y. The derivative of e to the y is e

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to the y

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times the derivative of y, which is dy dx.

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The derivative of y cubed is 3y squared times dy over dx.

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3y squared times the derivative of y with respect x.

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And now if we want to find dy, dx, we have to solve for these.

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How are we going to do that?

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We got to bring both to one side.

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Okay. And dx, dx is?

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One.

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2 months, bud.

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Okay, try them both to one side.

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There is the implicit derivative. It is not pretty, but it gives a tangent slope at

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any point on the curve.

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But if we note any ordered pair that's on the curve,

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we plug in the x-coordinates into the x's, the y-coordinates into the y's,

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and we're going to get some slope of the tangent line.

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Okay, up that crazy looking curve.

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Okay, what questions we got on this?

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Differentiate as usual, and every time a variable depends on the differentiation

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variable, include the chain-rule derivative factor.

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by the chain will multiply by the derivative of that inside function with respect to

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any

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variable we want.

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Okay, if at the very beginning...

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If I said find

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dy, dt,

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instead of dy, dx,

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it would be the only thing we do different.

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The only thing different is

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And this would be DX DT.

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This would be DY DT.

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This would be DY DT.

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And then everything, this would not simplify to one.

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It just stay in there part of that product.

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Makes sense?

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Kind of?

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OK.

00:25:46.960 --> 00:25:48.680
Let us do another problem.

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What is this formula for?

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The volume of the classroom here?

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The volume of a cylinder is the area of the circular base times its height.

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This is the area of the base times our hives.

00:26:23.940 --> 00:26:26.640
Here volume is a function of what?

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Volume is a function of both radius and height.

00:26:37.500 --> 00:26:38.880
Do we see any X's?

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What is the variable? Radius and height? Yeah, here the volume is a function of both

00:26:44.152 --> 00:26:45.800
the radius and the height.

00:26:53.800 --> 00:27:04.840
We could find dV over dr, dV over dh, or dV over dt.

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We will find dV over dt, so volume, radius, and height are all functions of time.

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and A's are functions of T's. Okay, so let's start the left-hand side. We

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differentiate this with

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respect to time and it is or with respect to T. That was the left side easy. Now the

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right-hand

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hand side. What do we have?

00:27:53.460 --> 00:27:57.220
A product. We do have two products, but it's

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just a constant. So, let's keep it with this guy right here.

00:28:04.580 --> 00:28:05.120
Okay,

00:28:05.120 --> 00:28:06.280
that's that product right there.

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So, we need what role?

00:28:16.120 --> 00:28:17.280
Okay, so.

00:28:19.020 --> 00:28:21.700
The derivative is dV over dt.

00:28:26.020 --> 00:28:34.780
The derivative of pi r squared is 2 pi r times dr over dt.

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Then apply the product rule and include dh over dt on the second term.

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And that's it. Okay. Why would we ever want to do this?

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So soon we're gonna have cylinders pretend this is a cylinder.

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It does not.

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Water is not coming out of it.

00:29:35.160 --> 00:29:36.440
Water is going in. Water is going in.

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Oh!

00:29:38.520 --> 00:29:40.100
But it does have a leak.

00:29:40.380 --> 00:29:41.300
So it's going out as well.

00:29:42.380 --> 00:29:43.840
Guys, why are you freaking out?

00:29:45.180 --> 00:29:46.260
This is it right here.

00:29:49.780 --> 00:29:51.300
This is how...

00:29:51.300 --> 00:29:54.740
This is the rate of change of volume with respect to time.

00:29:56.020 --> 00:30:04.020
These factors describe how radius and height change with time.

00:30:05.140 --> 00:30:08.920
And they're all related. Why are they all related? Or how are they all related?

00:30:11.940 --> 00:30:14.460
They're related by exactly this equation.

00:30:17.420 --> 00:30:19.660
Okay, this is the relation right here.

00:30:20.420 --> 00:30:26.443
So if I know how fast I'm pouring water in, that would be the rate of change of the

00:30:26.443 --> 00:30:27.040
volume.

00:30:27.040 --> 00:30:33.720
Let's say I'm putting water in at three gallons an hour.

00:30:34.100 --> 00:30:35.060
It's pretty slow, but who cares?

00:30:38.100 --> 00:30:41.200
This would be our dv dt.

00:30:44.320 --> 00:30:45.220
Okay.

00:30:46.500 --> 00:30:54.971
And so for example if the radius is not changing right, fix cop the radius doesn't

00:30:54.971 --> 00:30:55.721
change.

00:30:56.280 --> 00:31:04.939
So would be the rate of change of the radius would be DR. That'd be zero. Okay the

00:31:04.939 --> 00:31:05.420
height

00:31:05.420 --> 00:31:12.864
would be changing as we add water and it's changing exactly this whatever that is.

00:31:12.864 --> 00:31:13.360
So

00:31:13.360 --> 00:31:15.580
If we plug in three actually right here,

00:31:17.440 --> 00:31:19.620
and we knew the radius of the top, whatever,

00:31:20.180 --> 00:31:22.880
and we could find exactly how fast the height is changing.

00:31:26.020 --> 00:31:29.140
This topic is called related rates.

00:31:31.600 --> 00:31:33.260
It is super, it was easy.

00:31:33.800 --> 00:31:34.700
We just did it.

00:31:38.180 --> 00:31:41.560
This is exactly how the different rates are related.

00:31:41.560 --> 00:31:44.393
the rate of change of the volume the rate of change the radius and the rate of

00:31:44.393 --> 00:31:45.143
change of the height.

00:31:45.540 --> 00:31:49.040
It's not complicated. It's that easy. Okay.

00:31:54.740 --> 00:32:01.620
What if, instead of dV over dt, we find dV over dh?

00:32:04.400 --> 00:32:05.960
What would be the only difference?

00:32:09.240 --> 00:32:14.140
We have an 8 here, an 8 here, and an 8 here.

00:32:15.060 --> 00:32:16.420
And then this would simplify to 1.

00:32:17.700 --> 00:32:19.260
Well, it's not really simplifying to d1.

00:32:19.820 --> 00:32:21.420
That's easy peasy.

00:32:25.120 --> 00:32:26.820
And then we're done.

00:32:30.000 --> 00:32:31.920
This is why I like the fractions.

00:32:33.760 --> 00:32:41.781
Okay, what does R prime even mean? What does H prime mean? The ray of change, but

00:32:41.781 --> 00:32:44.140
with respect to what variables?

00:32:47.040 --> 00:32:58.140
But how do we know that? How do I know that H prime is equal to 1?

00:33:00.220 --> 00:33:06.230
Why is x prime equal to 1? Only when we're differentiating with respect to x. Only

00:33:06.230 --> 00:33:08.860
when we're differentiating with respect to h.

00:33:09.360 --> 00:33:22.394
Leibniz notation records the variable with respect to which each derivative is

00:33:22.394 --> 00:33:23.480
taken.

00:33:26.540 --> 00:33:32.832
Implicit differentiation uses the same derivative rules, with a chain-rule factor

00:33:32.832 --> 00:33:35.120
for each dependent variable.

00:33:35.120 --> 00:33:39.426
derivative of the function you multiply by the derivative of the variable with

00:33:39.426 --> 00:33:40.420
respect to whatever

00:33:40.420 --> 00:33:41.320
favorite.

00:33:42.520 --> 00:33:43.900
Can you try one more?

00:33:44.660 --> 00:33:45.560
Where should we start at home?

00:33:50.020 --> 00:33:50.920
One more.

00:33:51.200 --> 00:33:52.100
Let's do one more.

00:33:52.220 --> 00:33:53.120
I like it.

00:33:54.860 --> 00:33:57.740
We will find dx over dn.

00:34:00.680 --> 00:34:01.860
Whatever that means.

00:34:04.320 --> 00:34:09.060
It's the rate of change of X with respect to the change in n.

00:34:11.140 --> 00:34:12.400
So we started the left hand side.

00:34:13.940 --> 00:34:14.880
The left-hand side requires the quotient rule.

00:34:16.180 --> 00:34:17.220
Quotient rule.

00:34:17.580 --> 00:34:18.560
We've got to use the quotient rule.

00:34:30.600 --> 00:34:31.560
So, the

00:34:31.560 --> 00:34:33.640
The derivative of x to the fourth is 4x cubed times dx over dn.

00:34:36.340 --> 00:34:37.700
4x cubed.

00:34:37.700 --> 00:34:40.240
Times the derivative of our inside,

00:34:40.760 --> 00:34:41.880
which is

00:34:43.960 --> 00:34:47.160
dx with respect to n.

00:34:47.900 --> 00:34:49.140
We don't care what goes.

00:34:52.820 --> 00:34:54.960
And then the second function

00:34:57.660 --> 00:34:59.680
minus the first function.

00:35:00.780 --> 00:35:04.320
Differentiate square root of y as one over 2 square root of y times dy over dn.

00:35:08.560 --> 00:35:22.037
y to the negative one half times the derivative of y which is dy dn divided by the

00:35:22.037 --> 00:35:25.780
square root of y squared.

00:35:30.440 --> 00:35:39.840
The right-hand side is a product.

00:35:39.840 --> 00:35:41.780
So we got to use the product pool.

00:35:50.240 --> 00:35:57.760
The derivative of x cubed is 3x squared times dx over dn.

00:35:59.420 --> 00:36:00.460
dx dn.

00:36:02.600 --> 00:36:05.980
Then keep x cubed and differentiate y as dy over dn.

00:36:08.100 --> 00:36:13.860
So, dy over dn.

00:36:14.360 --> 00:36:18.149
You could think of it as the derivative of y as 1, but then we still have to

00:36:18.149 --> 00:36:18.360
multiply

00:36:18.360 --> 00:36:22.920
by the derivative of that variable, which is dy over dn.

00:36:26.300 --> 00:36:27.200
Okay.

00:36:28.980 --> 00:36:30.440
If I said find

00:36:30.440 --> 00:36:34.220
dy dx, what would be the only change?

00:36:37.580 --> 00:36:39.140
All the dn's would be

00:36:39.140 --> 00:36:42.920
dx's and then the dx over dx

00:36:42.920 --> 00:36:43.980
simplifies to one.

00:36:46.460 --> 00:36:50.020
What was it? I asked for dx dn.

00:36:54.100 --> 00:37:00.942
We could collect all terms containing dx over dn and solve, but we will stop after

00:37:00.942 --> 00:37:03.080
setting up the derivative correctly.

00:37:03.740 --> 00:37:09.290
We'd have to clear the denominator here, bring all everything with the DX on one

00:37:09.290 --> 00:37:12.620
side, everything without DX, XDN on the other side.

00:37:13.100 --> 00:37:17.160
But we're going to stop there. That wasn't bad, right?