Witch of Agnesi Let y(x²+4)=8 (see figure).

Question

d. Verify that the results of parts (a) and (c) are consistent.

Accepted Answer

Hello. In this video, we are going to be solving the following implicit differentiation problem. We are asked to determine the derivative DYDX for Y multiplied by X2 + 36 equal to 216 using implicit differentiation. We will then verify the results by finding an explicit expression for Y and then differentiating to obtain DYDX. So, let's go ahead and work on the first part of this problem. The first part of this problem is asking us to solve for DYDX for the given equation by using implicit differentiation. Now, the equation given to us is Y multiplied by the quantity X2 + 36 equal to 216. In order to solve for DYDX, we will need to take the derivative of the entire equation with respect to X. Now, in order to take the derivative of the left hand side, we are going to need to use the product rule, since we have a function of I of Y multiplied by a function of X. That is going to give us Y multiplied by the derivative of X2 + 36, which is going to be 2 X by the power rule. Then we will add X2 + 36. Multiplied by the derivative of Y. Now, the derivative of Y is going to be one by the power rule, but since we are taking the derivative of Y with respect to X, that is going to generate a DYDX term, and one multiplied by DYDX is going to leave us with DYDX. So this will be the derivative of the left hand side of the equation. On the right hand side, we will take the derivative of 216, and that is going to be zero, because the derivative of a constant will always be zero. Now what we want to do is we want to solve for DYDX. If we multiply Y and two X together, that is going to simplify to give us 2 XY. We will subtract it 2 XY to both sides of the equation, leaving us with the quantity X2 + 36 multiplied by DYDX equal to negative to XY. And if we divide both sides of this equation by X2 + 36, DYDX will equal to -2 XY divided by X2 + 36. This is going to be the solution to the first part of the problem. This is the derivative DYDX. Now what we want to do is we want to work on the second part of the problem. We are going to verify this result by solving for an explicit expression for Y. Then we will differentiate that expression to obtain D Y D X. So, going back to the original problem, the original problem or the original equation is Y multiplied by X2 + 36, equal to 216. So, what the second part of the question is asking us to do is to solve for Y from the original equation. In order to do this, we will divide both sides of the equation by X2 + 36, and that will give us Y equal to 216 divided by X2 + 36. Now what we want to do is we want to take the derivative of this expression with respect to X. That is going to give us DYDX equal to X2 + 36 multiplied by 0 minus 216 multiplied by 2 X all divided. By X2 + 36 quantity squared. This is going to be the derivative via the quotient rule, and now we need to simplify. X2 + 36, multiplied by 0, will cancel to be 0. And -216 multiplied by 2 X will simplify to be negative 400. -432. So the derivative DYDX. Of the expression for Y simplifies to -432 X divided by the quantity X2 + 36 quantity squared. So this is going to be the derivative. Of the expression of why. And we want to verify that this is the derivative by using the expression of why we solved for. And plugging it into the implicit derivative. By taking this value of Y and plugging it into the implicit derivative, we get D Y D X equal to -2 X multiplied by 216 divided by X2 + 36, and this will all be divided by X2 + 36. We will multiply -2 X and 216 divided by X2 + 36 in the numerator. That will simplify the numerator to be negative 432 X divided by X2 + 36, and this will all be divided by X2 + 36. Finally, by simplifying the complex fraction, we can simplify this DYDX derivative. To be negative 432 X divided by the quantity X2 + 36 squad. And this verifies that our original derivative, or the implicit derivative of Y is the overall derivative for DYDX. And with that being said, the overall solution. Is going to be D. So I hope this video helps you in understanding how to verify an implicit derivative, and I will go ahead and see you all in the next video.

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
d. Verify that the results of parts (a) and (c) are consistent.

Key Concepts

Witch of Agnesi

Verification of Results

Graphical Interpretation

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