If r + s² + v³ = 12, dr/dt = 4, and ds/dt = –3, find dv/dt whe...

Question

If r + s² + v³ = 12, dr/dt = 4, and ds/dt = –3, find dv/dt when r = 3 and s = 1.

Accepted Answer

Welcome back, everyone. If A plus 3B squad minus 2 CQ equals 3, D A divided by DT equals -1, and DB divided by DT equals 4, DC divided by DT1A. equals 2 and B equals 1. A says 29 divided by 6. B 6 divided by 23, C-23 divided by 6, and D 23 divided by 6. So basically we're given the derivatives of A, B, and C. What we can conclude is that A is a function of T, B is also a function of T, and C is a function of T. What we're going to do is basically perform implicit differentiation for the given expression. First of all, we have the derivative of A, which is the A divided by DT. Plus, we need to differentiate 3 B squared according to the power rule, that's 2 B. We have to first of all differentiate the outer function and then apply the chain rule, right? B is a function of T, so we're also multiplying by the B divided by the T. Then we are subtracting 2 multiplied by the derivative of C cubed according to the power rule that's 3C2d and because C is a function of T we multiply by DC divided by DT. On the right hand side, the derivative of 3 is 0 because 3 is a constant. So from here, we can just substitute what is given. We know that the divided by DT is -1. We get plus 3 multiplied by 2 multiplied by B. The value of B is given as 1, multiplied by DB divided by DT, which is 4, and we are subtracting 2 multiplied by 3 multiplied by C2, so C is not given to us, we're going to leave it as C2 multiplied by the C. Divided by the T. This is going to be equal to 0. From here, let's simplify. We're going to get -1 plus 3 multiplied by 2 multiplied by 4. This is 24. We are then subtracting 6 C2 DC divided by DC. Which is equal to 0, and now we can just rearrange the expression, right? We can move negative C2 DC divided by DT to the right, which gives us positive C2 DC divided by DT equals to -1 + 24, which is 23. And now dividing both sides by 6. We get C2 DC divided by DT equals 23 divided by 6. So to find DC divided by DT we need to know the value of C2d and to identify the value of C, we're going to use the original expression, right? We can identify that variable knowing that A is equal to 2, rights, we're going to substitute A equals 2. We're adding 3 B squad, so 3 multiplied by 1 squared. We're subtracting 2 C cubed, C is unknown. And that must be equal to 3. Now, let's simplify. We got 2 + 3. Which is 8. I'm sorry, 2 + 3, which is 5. So let's fix that. We have 5 minus 2 CQ is equal to 3. Now let's rearrange. We have 2 C cubed equals 3 minus 5, which is -2, and this tells us that C cubed must be equal to 1, meaning C is equal to 1. And now that we know that C is equal to 1, we can show that the C divided by the T multiplied by C2, which is 1 squared or just 1. Is equals to 23 divided by 6. So DC divided by DT is equal to 23 divided by 6, which corresponds to the answer choice D. Thank you for watching.

If r + s² + v³ = 12, dr/dt = 4, and ds/dt = –3, find dv/dt when r = 3 and s = 1.

Key Concepts

Implicit Differentiation

Chain Rule

Substitution of Known Values

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