Differentiating Implicitly

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Question

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x²(x – y)² = x² – y²

Accepted Answer

Hello there. Today we're going to 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. Use implicit differentiation to find Dy by DX for the equation, parentheses X plus to the power of 2 is equal to x2 + Y2. Awesome. So it appears for this particular problem we're asked to use implicit differentiation to help us find Dy by DX for our provided equation by the prom itself. So now that we know we're ultimately trying to find DY by DX, let's read off our multiple choice answers to see what our final answer might be. A is negative XY, B is XY, C is negative Y divided by X, and D is Y divided by X. Awesome. So first off, in order to find Dy by DX using implicit differentiation for the equation, let us quickly once again write down our given equation to know what equation we're trying to find or use implicit differentiation on. So let us know that we're told that parentheses X plus Y to the power 2 is equal to x2 + Y2. So in order to help us To solve for our implicit differentiation, our first major step towards solving this problem is we need to differentiate both sides of our expression. And let us start with the left hand side. So let us start with D by DX of X + Y to the 2, so parentheses X + Y to the power 2. So in order to do this particular differentiation, we need to recall and The chain rule, so when you take D by DX of parentheses X plus Y to the power of 2, which will be equal to 2 multiplied by X + Y multiplied by D by DX of X + Y. Fantastic. So now we need to differentiate X plus Y with respect to X. So when we do that, we will find that D D X of X + Y will be equal to 1 + DY by DX. So that will mean that the derivative of the left hand side will be Drum roll, 2 multiplied by X + Y in parenthesis multiplied by 1 plus Dy by DX. Fantastic. So now we need to differentiate the right hand side, so we need to take D by DX of X2 + Y2, which will be equal to 2 X plus 2y. Multiplied by Dy by DX. So now our second step is we need to set up our equation. So we need to note that we now have 2 multiplied by X + Y in parentheses multiplied by 1 + Dy by D X is equal to 2 X plus 2y. Multiplied by Dy by DX. So now our third step is we need to solve for Dy by DX and in order to do that, we must first distribute on the left side, so we need to go ahead and write that 2 multiplied by parentheses X + Y plus 2 multiplied by parenthesis of X + Y multiplied by Dy by DX. is equal to 2 X plus 2y. So when you take 2 X plus 2y multiplied by Dy by DX, fantastic. So now we need to collect terms involving DY by DX onto and we're trying to move it to solve for Dy by DX on one side. So we're trying to move Dy by DX to one side since we're trying to solve for DY by DX ultimately. So with that in mind, we need to go ahead and write that. 2 multiplied by parentheses X plus Y minus 2 X is equal to 2. Y multiplied by Dy by D X minus 2 multiplied by parentheses X plus Y multiplied by Dy by DX. So when you take D Y by DX, fantastic. So now we need to factor out DY by DX on the right hand side. So when we do that, we can go ahead and write that 2 multiplied by parentheses X plus Y minus 2 X is going to be equal to Dy by DX so it's gonna be D by DX. Multiplied by 2 Y minus 2 multiplied by parentheses X plus Y. Fantastic. So now we need to simplify the terms within our parentheses, so we need to take 2 multiplied by parentheses X plus Y minus 2 X is equal to Dy by DX multiplied by 2Y minus 2X minus 2Y. And when we simplify again, we'll get 2 multiplied by parentheses X plus Y minus 2 X is equal to DY by DX multiplied by -2 X, so -2 X fantastic. So now, our 4th step. That we're on to now. So our fourth step is we now need to solve for D Y by DX. So we need to take 2 multiplied by X plus Y in parentheses. Minus 2 X is equal to -2 X multiplied by Dy by DX. So now all we need to do is divide both sides by -2 X to get Dy by DX all by itself, and when we do that, we will find that Dy by DX is going to be ultimately equal to when we simplify down to its most simple form, I'm gonna skip some steps. So when we divide -2 X to the other side, and we simplify. Ultimately, we will find that DY by DX when I skip a few steps, will be ultimately equal to negative Y divided by X. And that's it. That is our final answer in its most simple form. So once again, our final answer without a doubt has to be DY by DX is equal to negative Y divided by X. Fantastic. So going back to look at our multiple choice answers, the correct and final answer, without a doubt, will have to be What appears to be multiple choice answer letter C. So once again, C once again is negative Y divided by X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x²(x – y)² = x² – y²

Key Concepts

Implicit Differentiation

Chain Rule

Product Rule

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