Computer Explorations

Use a ...

Question

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.

2y² + (xy)¹/³ = x² + 2, P(1,1)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find Dy by DX 432 + 4 Xy the power of 1/3 is equal to x 2 + 1 and evaluate it at the parentheses 21. Awesome. So it appears for this particular problem, we're asked to solve for two separate answers. Our first answer that we're trying to solve for is we're trying to find the value of DY by DX. For our specific expression or function that is given to us. So first answer, once again, we're trying to find DY by DX ultimately. And our second answer is once we determine the value for Dy by DX, we're trying to evaluate it at the parentheses 2.1. So that's our second answer we're trying to solve for. We're trying to take our value for DY by DX and we're evaluating it at the point 2.1 in parentheses. Awesome. So now that we know what we're ultimately trying to solve for, what we need to do first off is we need to differentiate both sides of our function with respect to X. So we need to take 6 Y multiplied by Dy by DX plus 1/3 multiplied by parentheses 4 XY. To the power of negative 2/3, multiplied by 4 XY to the power of prime is equal to 2 x + 0. And we can go ahead and take it a step further and write 6 Y. 6 Y multiplied by Dy by DX plus 1/3 multiplied by parentheses 4 XY to the power of negative 2/3 multiplied by 4 X multiplied by Dy by DX plus 4 Y. Is equal to 2 X. We can also go ahead and write that 6 Y multiplied by DY by DX plus 4/3. X multiplied by 4 XY to the power of negative 2/3 multiplied by Dy by DX plus 4/3 or 4/3 multiplied by Y multiplied by parenthesis, 4 XY to the power of negative 2/3 is equal to 2 X. We can take it a step further and we can also write parentheses 6 Y plus 4/3. Multiplied by X, multiplied by parentheses 4 XY to the power of -23 multiplied by Dy by DX is equal to 2 X minus 4/3 multiplied by Y multiplied by parentheses 4 XY to the power of -2/3. Fantastic. So now rearranging our equation to isolate and solve for Dy by DX by all by itself on one side because that's what we're ultimately trying to solve for is we're trying to isolate and solve for DY by DX. And when we do that, we will find that DY by DX is equal to 2X minus 4. Thirds multiplied by Y multiplied by 4 XY to the power of negative 2/3. Fantastic, all divided by 6 Y plus 4/3 multiplied by X, multiplied by 4 XY to the power of negative 2/3. However, even though we've officially isolated and solved for DY by DX all by itself on one side, this is not our final answer PS right now, as our answer is for DY by DX. It is not in its most simplified form. So you need to keep simplifying canceling out like terms and simplifying it down to the point where you can't simplify any further, and that will be technically our final answer. So, In an effort to save some time, I'm going to skip a few steps, but like I said, we're trying to rearrange this expression to simplify it down to its most simple form. And when we do that, we will find that our final answer will have to be, when we've done our simplification process correctly, it'll be equal to 3 X multiplied by 4 XY. To the power of 2/3 minus 2y all divided by 9 Y multiplied by 4 XY to the power of 2/3 plus 2 X. And that's it. That is our first final answer that we're trying to solve for. Hooray, we did it. So now we need to solve for our second answer that we're trying to solve for. So what we're trying to do now is we're trying to substitute our point 2.1 into our expression for DY by DX. So with that said, we need to take DY by DX and we need to evaluate it at 2.1. So when we do that, we will find that it's equal to 3 multiplied by 2, multiplied by 4 multiplied by 2, multiplied by 1. To the power of 2/3 minus 2 multiplied by 1, all divided by 9 multiplied by 1, multiplied by 4 multiplied by 2, multiplied by And so it's gonna be 4 multiplied by 2, multiplied by 1, all to the power of 2/3 plus 2 multiplied by 2. So when we plug in this big old long expression into our calculator and we keep it in fraction form, we will find that our final answer will have to be equal to 11 divided by 20 or 11/20. And that's it. Those are our final two answers. Hooray, we did it. So looking at the problem itself and quickly recapping everything that we've learned together, our final two answers, without a doubt, have to be once again. D Y D X is equal to 3 x multiplied by parentheses 4 xy to the power of 2/3 minus 2y all divided by 9y multiplied by parentheses 4 Xy to the power of 2/3 plus 2 X. And DY by DX evaluated at the parentheses 21 will be equal to 11 divided by 20 or 11/20. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.

2y² + (xy)¹/³ = x² + 2, P(1,1)

Key Concepts

Implicit Differentiation

Chain Rule

Evaluating Derivatives at a Point

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