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

Question

c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.

Accepted Answer

Hello. In this video, we are going to be approaching the following derivative problem. We are asked to find an explicit expression for Y for the following equation Y multiplied by X2 + 16 equal to 64. Then we want to determine the derivative DYDX. So, let's go ahead and start by approaching the first part of this problem. We want to solve for an explicit expression for Y. Now, the given expression is Y multiplied by X2 + 16, equal to 64. The first part is asking us to solve for the variable Y. Now, since Y is part of this product on the left-hand side of the equation, all we need to do is divide by X + 16. That will give us Y is equal to 64 divided by the quantity X2 + 16. This is going to be the explicit expression for the for the variable Y. Now, the second part of this problem is asking us to determine the derivative of DYDX. In order to do this, we will take our expression of Y, then take the derivative with respect to X. So, let's go ahead and take the derivative of the left hand side of the equation. In order to take the derivative of Y, we will use the power rule. The derivative of Y is going to be 1, but since we took the derivative of Y with respect to X, that is going to generate a DYDX term through implicit differentiation. One, multiplied by DYDX is going to give us DYDX. So that is the derivative of the left hand side of the equation. But what about the derivative of the right hand side? Well, in order to take this derivative, we are going to have to use the quotient rule, since we are taking the derivative of a rational function. Let's go ahead and set the numerator as an F of X function. We will set that as F of X equal to 64, and we will set the denominator G of X. As X2 + 16. What we want to do now is you want to take the derivatives with respect to X. The derivative of a constant 64 will be 0, so F of X is equal to 0. And the derivative G of X with respect to X is going to be 2 X by the power rule, and the derivative of 16 is just 0, so G of X is equal to 2 X. Now that we have the pieces of information we need for the quotient rule, let's go ahead and plug it into the quotient rule in order to find the derivative of the right hand side. That is going to be G of X, which is X2 + 16, multiplied by the derivative of F of X, which is 0, minus F of X 64, multiplied by the derivative of G of X, which is 2 X, all divided by the original G of X2, which is going to be X2 + 16 squad. Now what we need to do is we need to simplify the derivative. X2 + 16, multiplied by 0, is just going to be 0. And -64, multiplied by 2 X is going to give us -128 X. So the derivative DYDX is equal to -128 X divided. By X2 + 16 quantity squared. And this is going to be the derivative. Of the expression for why. And with that being said, the overall solution is going to be D. So, I hope this video helps you in understanding on how to approach this type of problem, and I will go ahead and see you all in the next video.

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.

Key Concepts

Implicit Function

Differentiation

Witch of Agnesi

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