If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when ...

Question

If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.

Accepted Answer

Welcome back, everyone. Given a curve defined by the equation, S E equals the square root of U2 plus V2, where the derivative of U with respect to T equals 2 and the derivative of V with respect to T equals -4, calculate the derivative of S with respect to T when U equals 8 and V equals 6. A says it's 4/5, B 45, C 5/4, and D 54. Now, we're trying to calculate the derivative of S with respect to T, and we have some information that will probably help us later on down the road. So let's first differentiate S with respect to T and then see how we can use those pieces of information in the problem statement. Now we know that S is equal to the square root of u2 plus V2, which we could rewrite as u2 plus V2 raised to the power of a half and no notice that it is a composite function in which we'll be able to differentiate using the chain root. And know by the chain rule of differentiation, basically, what it means is that the derivative of S with respect to T is going to be equal to the derivative of the alter function in our uh composite function here. So to differentiate that, we bring down the power half right by the function U2 plus V2d and subtract 1 from the power. Uh, so that's gonna be negative or half, OK, multiplied by the derivative of the inner function. In other words, the derivative of U squared plus V2, which is gonna be 2 UU DT, OK, plus 2 V DVDT by implicit differentiation. Now, if we simplify this expression here, OK, then we know that first, we can factor 02 from the term 2UDUDT plus 2 VDVDT, OK? So that's gonna be a half multiplied by 1 divided by the square root of U2 plus V2d, OK, multiplied by 2, multiplied by UD UDT plus VDVDT. And no, we can cancel 2 or multiply 2 by 2. It gives us 1. And this is gonna simplify to UD UDT. Plus VDVDT. All divided by the square root of U2 plus V2. Now remember, we know the values of U, V, DU DT and DVDT. So if we substitute those, we know U equals 8, DUT equals 2. V equals 6, OK, and DVDT equals -4. And all of this will be divided by the square root of U2, so that's 82 plus V2, which is 62. No, 8 multiplied by 2 is 16. OK, 6 multiplied by -4 is 24, 8 squared is 64, and 6 squared is 36, and 64 plus 36 equals 100. So in our denominator, it would be the root of 100. No 16 minus 24 is -8, and the square root of 100 is 10, so that means the derivative of S with respect to T equals -8/ten or simplified negative4/5. If we look back at our answer choices, that means B is the correct option. Thanks a lot for watching everyone. I hope this video helped.

If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.

Key Concepts

Chain Rule

Implicit Differentiation

Pythagorean Theorem

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