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Suppose $ 4x^2 + 9y^2 = 36, $ where $ x $ and $ y $ are functions of $ t $.(a) If $ dy/dt = \frac {1}{3}, $ find $ dx/dt $ when $ x = 2 $ and $ y = \frac {2}{3} \sqrt 5. $(b) If $ dx/dt = 3, $ find $ dy/dt $ when $ x = -2 $ and $ y = \frac {2}{3} \sqrt 5. $

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00:33

Alex Lee

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 9

Related Rates

Derivatives

Differentiation

Campbell University

Oregon State University

Baylor University

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

0:00

Suppose $ y = \sqrt {2x + …

01:36

04:19

Supposey =square r…

02:36

Suppose $4 x^{2}+9 y^{2}=3…

03:13

02:13

If $ x^2 + y^2 + z^2 = 9, …

00:42

(a) Find the differential …

02:21

Suppose $4 x^{2}+9 y^{2}=2…

04:09

Suppose $y=\sqrt{2 x+1},$ …

02:09

02:48

Find the differential $d y…

01:21

Compute $dy/dx$ for the fo…

Why did you say we're told that four X squared plus nine y squared equals 36 on the show? So this is the equation of an ellipse where X and y are both functions of t Get this. Pardon me. We're told that dy DT equals one third and ratifying the x tt when X equals two. Yes. Cigarette. Yeah. And why equals two thirds of route five? Yes, disappear. So in order to find the x c t, I'm going to use implicit differentiation on our lips equation. So differentiating both sides of respect the teeth. So we have two times four is eight x times The expertise plus two times nine is 18 times y times dy DT and the right hand side is simply zero solving for the execution, the expert C is equal to negative 18 y times dy DT over eight x which you could simplify to negative nine wide dy DT over guys, we're we're at X plugging in our values. You get negative nine times two thirds route five times one third over four times two it's ms is equal to negative. See the 9/9 minutes to five or four times two, which is Route 5/4. All right, please. This is DX DT Negative. Route 5/4, actually, Sharp party. We're told that the X 15 equals three and rectifying the white tea when X equals negative two. And why it was two thirds or five. Said was plan now, just like in part a use implicit differentiation. And we get the same equation and I want to solve for dy DT instead of the expertise. So we have D y t t is equal to negative eight x the x 30 over 18 y couldn't you just lie up? This simplifies to negative four x over nine y the x 30 substituting our values. This is native four times negative. Two over nine times two thirds Route five Nice three is equal to the negatives. Cancel out 8%. Now this is eight over 19, 0 to 5, which is 4/5. And if you rationalize the denominator, this is equal to four. Route 5/5. Oh,

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