Use implicit differentiation to find dy/dx.
x³

Question

Use implicit differentiation to find dy/dx.

x³ = (x + y) / (x - y)

Accepted Answer

Hello. In this video, we are going to be calculating the following derivative. We are asked to find DYDX by implicit differentiation. The equation given to us is 7 X2 equal to 2 X minus Y divided by X + 2 Y. Now, we could just start by taking the derivative with respect to X, but before we do that, I would personally rewrite the equation in the form of a product, so that we can use the product rule instead of the quotient rule. What we can go ahead and do is multiply both sides of the equation by X + 2 Y. and that will allow us to rewrite the equation as X + 2 Y multiplied by 7 X2 equal to the quantity 2X minus Y. Then what we can do is we can distribute 7 X2 to the quantity X + 2 Y. That will allow us to rewrite the quantity as 7 X cubed plus 14 X 2 Y equal to 2 X minus Y. Now, what we want to do is we want to calculate the derivative DYDX. In order to do this, we are going to take the derivative of the entire equation with respect to X. So we will go ahead and take the derivative of each individual term. Starting with 7 X cubed, we will go ahead and take the derivative by using the power rule. That will give us 7 multiplied by 3 X squared. Now, for the second term on the left-hand side, notice that we have a product of X2 and Y. In order to take the derivative, we will need to use the product rule. So, we will take the coefficient of 14, and multiply this to the product rule of X2 multiplied by Y. That derivative will be X2d multiplied by the derivative of Y with respect to X. Now, the derivative of Y is going to be one by the power rule, but we are taking the derivative of Y with respect to X. So by implicit differentiation, that will be 1 multiplied by DY divided by DX. Then we will go ahead and add Y and multiply this by the derivative of X2, which is going to be 2 X by the power rule. Then we will take the derivative of the right hand side of the equation. The derivative of 2 X with respect to X is going to be 2 by the power rule. And again, the derivative of negative Y is going to be 1, but because we took the derivative of Y with respect to X, we will generate a DYDX term. Now what we need to do is we need to simplify. For the first term, 7 multiplied by 3 X2 will give us 21 X squad. X2 multiplied by 1, multiplied by DYDX is going to simplify to be X2 DYDX. And Y multiplied by 2 X is going to simplify to be 2 XY. Then we will distribute 14 to both of these terms. That is going to simplify to be 14. X 2 D Y D X Plus 28 XY. And that is going to equal to 2 minus DYDX. Now recall that we want to solve for DYDX. That is the goal of the problem. So what we can go ahead and do is we can move all the terms that have a DYDX factor to the right hand side of the equation and move all of the constant terms to the left hand side of the equation. So what we will do is we will subtract 14 X 2 DYTX to both sides of the equation, and subtract 2 to both sides of the equation. That will leave us with 21 X2 plus 28 XY minus 2 equal to -14 X2. Mine, uh DYDX. Minus DYDX. What we can go ahead and do is we can make all the DYDX terms positive by dividing every term in our equation by -1. If we divide everything by -1, and reorder the left hand side, that will give us 2 minus 21 X2 minus 28 XY equal to 14 X2DYDX plus DYDX. Now notice that both of the factors or both of the terms on the right-hand side of the equation, have a multiple of DYDX. In order to solve for DYDX, we can go ahead and factor out DYDX from each side of from each term on the right hand side of the equation. That will leave us with 2 minus 21 X2 minus 28 XY equal to the quantity 14 X2 plus 1 multiplied by DYDX. And finally, in order to solve for DYDX, we will divide both sides of this equation by 14 X2 + 1. That will give us the derivative DYDX equal to 2 minus 21 X2 minus 28 XY divided by the quantity 14 X2 + 1. And this is going to be the solution to the for the derivative of the problem. So, I hope this video helps you in understanding on how to use implicit differentiation, and I will go ahead and see you all in the next video.

Use implicit differentiation to find dy/dx.
x³ = (x + y) / (x - y)

Key Concepts

Implicit Differentiation

Chain Rule

Algebraic Manipulation

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