Find by implicit differentiation.
x² + xy + y² - 5x...

Question

Find by implicit differentiation.

x² + xy + y² - 5x = 2

Accepted Answer

Below 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. By using implicit differentiation, determine Dy by DX for the curve given by the equation 3 X2 minus 4 XY plus Y to the power of 3 equals 7. Awesome. So it appears for this particular problem we're ultimately trying to determine the value of Dy by DX for this curve that's given by our equation 3x2 minus 4 XY plus y to the power 3 equals 7 by using implicit differentiation. So now that we know what we're ultimately trying to solve for, we need to find Dy by DX. Our first step that we need to do in order to solve for so we're trying to find D by DX. So what we need to do first is we need to use implicit differentiation like it states, but we need to differentiate both sides of the equation with respect to X. So, let us differentiate each term individually. So, let us note that the derivative of 3 X to the power 2 with respect to X is going to be equal to 6 X. And for 4 XY, let us use the product rule. So when we use the product rule, we'll find that D by DX of -4 XY, it's gonna be equal to -4 multiplied by D by DX of X multiplied by Y. Plus X multiplied by Dy by DX. And when we do that, we will find that it's equal to -4, multiplied by Y plus X multiplied by DY by DX. Fantastic. And let us note that the derivative of Y to the power of 3 with respect to X is going to be done with using the chain rule. So we need to recall and use the chain rule to perform this differentiation. So D by DX of Y to the power of 3 is going to be equal to 3 Y 2 multiplied by Dy by DX. Fantastic. So now we need to differentiate the entire equation now, so we need to take D by DX 3 X to the power of 2 minus D by DX of 4 XY plus D by DX. So you take D by D X of the power of 3 is equal to D by DX of 7. And we need to substitute in our derivatives, so substituting in our derivatives, we will get 6 X minus 4 multiplied by Y plus X, multiplied by Dy by DX plus 3 Y 2 multiplied by Dy by DX is equal to 0. So now we need to simplify and solve for DY by DX, so we need to take 6 X minus 4 Y minus 4, X multiplied by Dy by DX plus 3Y to the power of 2 multiplied by Dy by DX is equal to 0. So we need to rearrange this expression to isolate and solve for DY by DX. So to do that we could take negative 4X multiplied by Dy by DX plus 3Y to the power 2 multiplied by Dy by DX. Fantastic. And we need to say that this is equal to 4 Y minus 6 X. So now we need to factor out DY by DX, so we need to take DY by DX, so we can say that DY by DX. Of -4 X plus 3 to the power of 2 is going to be equal to 4 Y minus 6 X. So now, when we get D Y by DX all by itself, you will find that DY by DX is going to be equal to 4 Y minus 6 X divided by 3 to the power of 2 minus 4 X, and that is our final answer. Hooray, we did it. So once again, our final answer without a doubt has to be DY by DX is equal to 4Y minus 6X, all divided by 32 minus 4X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find by implicit differentiation.
x² + xy + y² - 5x = 2

Key Concepts

Implicit Differentiation

Chain Rule

Collecting Like Terms

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