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.

x²/³ + y²/³ = 1

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 D2 Y by DX2 using implicit differentiation. X to the power of 3/4 plus the power of 3/4 is equal to 11. Awesome. So it appears for this particular prong where I'm trying to find the second derivative of Dy by DX using implicit differentiation, and we're trying to find the second derivative of Dy by DX of the equation that's given to us by the problem itself. So ultimately we're trying to find once again D2 Y by DX2, and that is our final answer we're ultimately trying to solve for. So with that said, first off, once again, in order to find D2 Y by DX2d using implicit differentiation for our equation, we need to note that we're trying to differentiate. X to the power of 3/4 plus Y to the power of 3/4 is equal to 11. And in order to solve for this particular problem, we need to split. Our differentiation into two different sides, and we need to differentiate both sides. So, let us start with the left hand side first. So we need to differentiate D by DX of X to the power of 3/4. Which will give us an answer of 3/4 multiplied by X to the power of -1/4. Fantastic. And then we need to take D by DX of Y to the power of 3/4, which is going to be equal to 3/4 multiplied by Y to the power of negative, 1/4 multiplied by DY by DX. Fantastic. So therefore, without a doubt, the derivative of the left hand side will be 3/4 multiplied by X to the power of negative 1/4 plus 3/4 multiplied by Y to the power of negative 1/4 multiplied by Dy by DX. Fantastic. So now we need to take the derivative of the right-hand side, so let us note that D by DX of 11, since it's taking the derivative of a constant value, is going to be equal to 0. So now, our next step is we need to set up our equation, so we need to take 3/4, so we take 3/4 X to the power of negative 1/4. Plus 3/4 multiplied by Y to the power of negative 1/4, multiplied by D Y D X is equal to 0, fantastic. So now, our next step is we need to solve for DY by DX. So DY by DX is going to be equal to negative. X to the power of 1/4 divided by y to the power of -14, which is going to give us an answer of when we evaluate negative y to the power of 1/4 divided by x to the power of 1/4. So now our next step is we need to differentiate again, because once again we're trying to find the second derivative of Dy by DX. So when we differentiate again, so once again we're trying to differentiate Dy by DX is equal to negative to the power of 1/4 divided by X to the power of 14th, and we can differentiate this expression by recalling and using the quotient rule. So we will find that D2 Y by DX2 is going to be equal to negative. 14th multiplied by Y to the power of -3/4 multiplied by Dy by DX multiplied by X to the power of 1/4 minus 1/4 multiplied by the power of 1/4. Multiplied by X to the power of -3/4, fantastic. And we need to divide everything by X to the power of 1/2. Fantastic. So now we need to substitute in the value of Dy by DX is equal to negative to the power of 1/4 divided by X the power of 1/4. So when we do that, we will find that D2 Y by DX2 will be equal to the power of -12 plus Y to the power of 1/4 multiplied by X to the power of -34. So X to the power of -3/4. All divided by 4 multiplied by X or 4 X to the power of 1/2. Fantastic. So now we need to solve even further. So when we solve even further, we will find that Y to the power of negative 12. Plus Y to the power of 1/4 multiplied by X to the power of 3/4, all divided by 4 X of 1/2. Awesome. So 4 x to the power of 12 will be equal to 1 divided by the power of 1/2. Plus, to the power of 1/4 divided by X to the power of 3/4. All divided by 4 X to the power of 1/2. Awesome. So ultimately, What we're trying to do is we're trying to simplify this expression down to its most simple form, which I'm in an effort to save time, I'm going to skip a few steps, but when we simplify our expression down to its most simple form, meaning we can't simplify it any further, we will find that it's going to be equal to X to the power of 3/4 plus Y to the power of 3/4. Divided by 4 X to the power of 5/4 multiplied by Y to the power of 1/2. And let us also note and recall once again that In our original problem, we're told that X to the power of 3/4 plus Y to the power of 3/4 is equal to 11. So, that will mean, without a doubt, our final answer will have to be drum roll, 11 divided by 4 X to the power of 5/4 multiplied by Y to the power of 1/2, and that's it. That is our final answer. Hooray, we did it. So going back to look at our original problem to quickly recap what we've learned, that will mean that our final answer, without a doubt has to be once again. 11 divided by 4 X to the power of 5/4 multiplied by Y to the power of 1/2. 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.

x²/³ + y²/³ = 1

Key Concepts

Implicit Differentiation

Chain Rule