A challenging derivative Find dy/dx, where √3x⁷+y² = sin²y+100...

Question

A challenging derivative Find dy/dx, where √3x⁷+y² = sin²y+100xy.

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. Find DY by DX for the equation, the square root of X2 + 8 multiplied by Y2. is equal to cosine squad of Y plus 72 multiplied by X multiplied by Y. Awesome. So it appears for this particular problem, we're ultimately trying to figure out Dy by DX for this specific equation. So now that we know that we're ultimately trying to find DY by DX for this specific equation, we need to recall and note that in order to find Dy by DX for this given equation, where our given equation once again is X2. Plus 8 multiplied by Y2 is equal to cosine squared of Y plus 72 multiplied by X multiplied by Y, we will need to recall and use implicit differentiation in order to perform this derivative. So, noting that we need to recall and use implicit differentiation, we need to differentiate both sides of this expression with respect to X. So let us start with the left hand side first. So firstly, focusing our attention on the left side of our expression first. So we need to differentiate. The square root of X2 + 8 Y2 with respect to X, so we need to take D by DX or D by DX is just a fancy way of saying we're taking the derivative with respect to X. So we're taking D by DX of the square root of X2 + 8Y2. Awesome. And this is equal to D by DX of X2 + 8Y2 to the power of 12, which is equal to Which we will say that this is all equal to 1 divided by 2 multiplied by the square root of X2 + 8 Y 2. Multiplied by 2 X. Plus 16 Y multiplied by DY by DX. Fantastic. So now, our next step is we need to simplify. So when we simplify, we will find that 1 divided by 2 multiplied by the square root of X2 + 8 Y2 multiplied by 2 X plus 16 Y multiplied by DY by DX, we will find that This will be all equal to when we simplify down to its most simple form. It'll be equal to X plus 8 Y multiplied by Dy by DX divided by the square root of X2. Plus 8 Y squared. Fantastic. So now we can now focus our attention on the right hand side. So now we're differentiating cosine squared of Y with respect to X, so we need to take D by DX of cosine squared of Y, which will be equal to 2 multiplied by cosine of Y multiplied by negative y. Multiplied by D Y by DX, which will be all equal to negative 2 multiplied by cosine of Y, multiplied by sin of Y. Multiplied by DY by DX. Fantastic. So now, our next step, our 3rd step is we need to differentiate. The value of 72 multiplied by X multiplied by Y with respect to X using the product rule, which, as we should recall, the product rule states that you multiplied by V is equal to V multiplied by U plus U multiplied by V. And we need to note that you will equal X in this case and V will equal Y in this case. So when we differentiate, we will find that D by DX of 72 multiplied by X multiplied by Y will be equal to 72 multiplied by Y plus X multiplied by Dy by DX, fantastic. So thus, therefore, the differentiated equation will be that X + 8 Y multiplied by DY by DX. Divided by the square root of X2 + 8 Y2 will be equal to -2 multiplied by cosine of Y multiplied by sin of Y. Multiplied by D by DX, so multiplied by DY by DX fantastic, plus 72, and we're gonna take 72, and we need to multiply it by Y plus X multiplied by DY by DX. Fantastic. So now, our next step at this point. Is we need to take. What we've learned so far, and we need to now solve for DY by DX in the differentiated equation. So in order to do that, our fourth step is we need to first clear the fraction by multiplying both sides by the square root of X2 + 8 Y2. and when we do that, we will find that We will get X + 8 Y. Multiplied by D Y D X is equal to negative 2 multiplied by cosine of Y multiplied by sin of Y multiplied by Dy by DX multiplied by the square root of X2 + 8 Y2. And we need to add 72 multiplied by Y + X multiplied by DY by DX. And we need to multiply. The square root of X2 + 8 Y2. So make sure that you need to multiply the square root of X2 + 8Y2. 272 multiplied by Y plus X multiplied by D Y by DX. So now, we need to distribute our square root of X2 + 8 Y2. And when we distribute, this is our 5th step, we will find that X plus So we need to take X plus 8 Y multiplied by Dy by DX is equal to -2 multiplied by cosine of Y multiplied by sin of Y. Multiplied by DY by DX. Multiplied by the square root of X2 + 8 Y2. Plus 72 multiplied by Y multiplied by the square root of X2 + 8 Y2 + 72 multiplied by X multiplied by D by DX multiplied by the square root of X2 + 8 Y2. Fantastic. So now, our next step now. we need to rearrange the terms in order to isolate DY by DX onto one side. So when we start to rearrange this expression, we will get 8 Y multiplied by DY by DX plus 2 multiplied by the cosine of Y multiplied by the sign of Y. Multiplied by DY by DX, multiplied by the square root of X2 + 8Y2. Minus 72 multiplied by X, multiplied by Dy by DX of, so it's gonna be multiplied by Dy by DX. Multiplied by The square root of X2 + 8 Y 2, and this will be all equal to 72 multiplied by Y multiplied by the square root of X2 + 8Y2. Minus X. So now our seventh step is we need to factor out DY by DX and when we do that, we will find that DY by DX. Multiplied by 8 Y plus 2 multiplied by the cosine of Y multiplied by the sign of Y multiplied by the square root of X2 + 8 Y2 minus 72 multiplied by X multiplied by the square root of X2 plus. 8 Y squad. will be all equal to 72 or 2 multiplied by Y multiplied by the square root of X2 + 8 Y2 minus X. So now our final step. is we need to factor out so we factored out D by DX. So now we can officially solve for DY by DX. So when we do that, we will find as our final answer that DY by DX will be equal to Drum roll, 72 multiplied by Y multiplied by the square root of X2 minus 8 Y2 minus X, all divided by. 8 Y plus 2 multiplied by cosine. Of Y multiplied by sign of Y. Multiplied by the square root of X2. Plus 8 Y squad. 72 X. And we need to take 72 X and we need to multiply it by the square root of X2 + 8Y2. And that is our final answer. Hooray, we did it. So, that will mean, without a doubt, so we ran out of room here, so let's move this down so we could see it a little bit better. So We need to note that once and for all, our final answer without a doubt will be that D Y by DX is equal to 72 multiplied by Y, multiplied by the square root of X2 minus 8 Y2 minus X all divided by. 8 Y plus 2 multiplied by cosine of Y multiplied by sin of Y multiplied by the square root of X2 + 8 Y2 minus 72 X multiplied by the square root of X2 + 8 Y2. And that's it. That is our final answer. Hooray, we did it. So therefore, we have officially solved for this problem. Awesome. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

A challenging derivative Find dy/dx, where √3x⁷+y² = sin²y+100xy.

Key Concepts

Implicit Differentiation

Chain Rule

Trigonometric Identities

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