Second Derivatives

In Exerci...

Question

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

y² – 2x = 1 – 2y

Accepted Answer

Hello there. Today we're gonna 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 the second derivative using implicit differentiation for the equation. 3 x 2 minus 2y 2 is equal to 6 XY minus 4. Awesome. So it appears for this particular problem we're asked to take our equation that is provided to us by the problem itself, and we're asked to find the second derivative by using implicit differentiation. So now that we know that we're ultimately trying to figure out what the second derivative is of our provided equation using implicit differentiation, let us first off note that we're told that the equation that we're going to be using to find the second derivative is 3X2 minus 2Y2 is equal to 6 XY. -4, and our first major step to start solving this particular problem is we need to differentiate implicitly with respect to X once, so we need to take 6. So when we're take so once again we're we're differentiating implicitly with respect to X. And when we do that, we will get 6 X minus 4 Y multiplied by Dy by DX which will be equal to 6 Y plus 6 X multiplied by Dy by DX. Fantastic. So now our next step is we need to solve for DY by DX, so we need to take 6 X minus 6Y is going to be equal to parentheses 6 X plus 4 Y multiplied by Dy by DX, which will be when we rearrange this expression to isolate and solve for DY by DX on one side, so Dy by DX. Is going to be equal to 6 multiplied by X minus Y divided by 2 multiplied by parentheses 3 X plus 2y. Which will be equal to, when we do a little bit of simplifying, which will be equal to 3 multiplied by parentheses X minus Y divided by 3 X plus 2 Y. Fantastic. So now our second step, now our second major step is we now need to differentiate again with respect to X. So we take D2 Y by DX2, which once again, this is just a fancy way of saying the second derivative of Dy by DX, which is equal to D by DX of. 3 multiplied by parentheses X minus Y divided by 3 X plus 2y. fantastic. So now, in order to perform this differentiation, we need to recall and use the quotient rule. So D squarey by DX2 is equal to parentheses 3. X plus 2 once again in parentheses, multiplied by parentheses 3 minus 3, multiplied by D by DX. Fantastic, minus 3 multiplied by X minus Y. Multiplied by parentheses 3 + 2 multiplied by Dy by DX. Fantastic. All divided by parentheses 3 X plus 2y all to the power of 2. Fantastic. So now, moving right along here, we now need to substitute in DY by DX and let's make a note of this in red. So right now, we need to note that we're taking DY by DX is equal to 3 multiplied by parentheses X minus Y divided by. 3 X plus 2Y, and we're gonna be taking this DY by DX expression, and we're gonna be substituting it in. So when we do that, we will find that it's gonna be equal to parentheses. 3 X plus 2y in parentheses multiplied by parentheses. 3 minus 9 multiplied by parentheses X minus Y, all divided by 3 X plus 2 Y minus. 3, so it would be minus 3, and then we're gonna take minus 3, and we're gonna multiply it by parentheses X minus Y. then we're gonna multiply it by 3, which is in parenthesis, we're going to take minus 3 multiplied by X minus Y multiplied by parentheses. 3 plus 6 multiplied by X minus Y. So X minus Y. Divided by 3 X plus 2Y, fantastic. And then this will be all divided by parenthesis, 3 X plus 2y to the power of 2. Awesome. So now, take a little breather. So now, our next step is we need to simplify our numerator. And I'm gonna skip a few steps, but using a little bit of algebra, you should get, when you simplify your numerator, you should find that this will be equal to -9 X plus 24, so -9X-9 X plus 24 Y minus 18. Multiplied by parentheses X minus Y to the power of 2, divided by 3 X plus 2y. And then what we're gonna do is we're gonna divide everything by. 3 X plus 2y to the power of 2. So once again, we simplified our numerator using a little bit of algebra. I skipped a few steps, but all you're trying to do is you're trying to simplify the numerator down to its most simple form where you can't simplify it anymore, and that's when you get the following answer. So now, our next step that we need to take together is we need to multiply the numerator and denominator by 3 X plus 2y. And when we do that, we will find that it's going to be equal to So once again, it's going to be equal to, so don't forget your equal signs. It's gonna be equal to, once again, just a quick reminder, we're multiplying the numerator and denominator by parentheses 3 X plus 2y. So when we do that, we will find that it's equal to parentheses -9 X plus 24, once again in parentheses, multiplied by parentheses 3. X + 2 Y minus 18 multiplied by parentheses X minus Y 2. Divided by 3 X plus 2y to the power of 3. Awesome. So now our next step. we need to expand. So once again, let's make a note that we're expanding parentheses 9 X plus 24, so -9 X plus 24Y in parentheses, multiplied by 3 X plus 2 Y. Awesome. So when we expand that, we will find that it's equal to, so -27X2 minus 18 XY plus 72. So it's gonna be plus 72 XY plus 40. 8, and don't forget that it's gonna be + 48 Y to the power 2, which is going to be equal to -27X2 + 54 XY. Plus, lastly, 48 Y squared. Fantastic. So now, moving right along here, we now need to expand. Drum rolls were expanding. 18 multiplied by, so we're going to take 18, multiplied by X minus Y to the power 2, and when we do that, we will find that it's equal to 18 multiplied by parentheses X2 minus 2 XY plus Y2, which is going to be equal to 18, and we're gonna take 18, and don't forget that it's 18 X2. Minus 36 XY plus 18 Y square. Fantastic. So now, We need to combine our like terms. So when we do that, we will find that it's gonna be -27 x 2 + 54 XY plus 48 Y2 minus 18 X2 + 36 XY minus 18 Y2, which is going to be equal to -45 X2. Plus 90 XY plus 30 Y squared. And when we factor this out, we will find that it's equal to 15 multiplied by parentheses 3 x 2 minus 6 X Y minus 2y 2. Fantastic. And let us note that from the original equation we can write that 3 x 2 minus 2y minus 6 X Y is equal to -4, thus we could go ahead and write that 3 x 2 minus 6 XY. Minus 2y quad is equal to -4. So, therefore, without a doubt, we could go ahead and write that. -15 multiplied by -4 is equal to 60. So therefore, without a doubt, D 2 Y by DX2, which is our second derivative for Dy by DX, is going to be equal to 60 divided by parentheses 3 X plus 2y to the power of 3, and that is our final answer. Hooray, we did it. So, going back to look at our pro itself to quickly recap what we've learned, our final answer once again, without a doubt, will have to be Once again, So, quickly recap, so our second derivative, once again using implicit differentiation, will be 60 divided by 3 X plus 2y in parenthesis to the power of 3. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

y² – 2x = 1 – 2y

Key Concepts

Implicit Differentiation

First Derivative (dy/dx)

Second Derivative (d²y/dx²)

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